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Practice: trig and inverse trig derivatives

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Wize University Calculus 1 Textbook > Derivatives

Derivatives of Trig Functions

6 Activities

Wize University Calculus 1 Textbook > Derivatives

Derivatives of Inverse Trig Functions

3 Activities

Wize University Calculus 1 Textbook > Derivatives

The Chain Rule

3 Activities

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Practice: trig and inverse trig derivatives

Find the derivative of the following trigonometric and inverse trigonometric functions:
a) f(x)=arcsin⁡(3x2)f(x)=\arcsin(3x^2)


b) y=(tan⁡2x)(arctan⁡(1−x))y=\left(\tan^2x\right)\left(\arctan\left(1-x\right)\right)

More Derivatives of Trig Functions Questions:
Quotient with Trig
Find the derivative of h(t)=cos⁡tsin⁡t(1+cos⁡t)\displaystyle h(t)=\frac{\cos t}{\sin t(1+\cos t)}
Practice: Equation of Tangent Line
Find the equation of the tangent line to the graph y=sin⁡x+3x2cos⁡xy=\sin x+3x^2\cos x at the point where x=π2x=\frac{\pi}{2}.
Practice: Equation of Tangent Line
Practice: Equation of Tangent Line
Find the equation of the tangent line to the graph y=sin⁡x+3x2cos⁡xy=\sin x+3x^2\cos x at the point where x=π2x=\frac{\pi}{2}.
Find the derivative of 𝑓(𝑥) = 𝑒𝑥 tan 𝑥 + 3.
Differentiation Rules
Find ddx(sin⁡2x+π+cos⁡2x+57)\frac{d}{dx}(\sin^2{x}+\sqrt\pi+\cos^2{x}+5^7)
Differentiation Rules
Find ddx(sin⁡2x+π+cos⁡2x+57)\frac{d}{dx}(\sin^2{x}+\sqrt\pi+\cos^2{x}+5^7)
Practice: Equation of Tangent Line
Practice: Equation of Tangent Line
Find the equation of the tangent line to the graph y=sin⁡x+3x2cos⁡xy=\sin x+3x^2\cos x at the point where x=π2x=\frac{\pi}{2}.
Practice: Equation of Tangent Line
Practice: Equation of Tangent Line
Find the equation of the tangent line to the graph y=sin⁡x+3x2cos⁡xy=\sin x+3x^2\cos x at the point where x=π2x=\frac{\pi}{2}.
Practice: Equation of Tangent Line
Practice: Equation of Tangent Line
Find the equation of the tangent line to the graph y=sin⁡x+3x2cos⁡xy=\sin x+3x^2\cos x at the point where x=π2x=\frac{\pi}{2}.
Derivatives: Trigonometric Functions
Find f′(π)+f′′(π)f'(\pi)+f''(\pi) if 𝑓(𝑥)=𝑥cos⁡𝑥−sin⁡𝑥𝑓(𝑥) = 𝑥 \cos 𝑥 − \sin 𝑥 .
Find the equation of the tangent line to the graph y=sin⁡x+3x2cos⁡xy=\sin x+3x^2\cos x at the point where x=π2x=\frac{\pi}{2}.
Practice: Equation of Tangent Line
Practice: Equation of Tangent Line
Find the equation of the tangent line to the graph y=sin⁡x+3x2cos⁡xy=\sin x+3x^2\cos x at the point where x=π2x=\frac{\pi}{2}.
Differentiation Rules
Find ddx(sin⁡2x+π+cos⁡2x+57)\frac{d}{dx}(\sin^2{x}+\sqrt\pi+\cos^2{x}+5^7)
Differentiation Rules
Find ddx(sin⁡2x+π+cos⁡2x+57)\frac{d}{dx}(\sin^2{x}+\sqrt\pi+\cos^2{x}+5^7)
Higher Order Trig Derivative
Find y23y^{23} given y=sin⁡x\displaystyle y=\sin x
Higher Order Derivatives
If f(x)=sin⁡(2x)f\left(x\right)=\sin\left(2x\right), find f(21)(π2)f^{\left(21\right)}\left(\frac{\pi}{2}\right).
(i.e. find the 21st derivative at the point π2\frac{\pi}{2})
Differentiating functions
Find the derivative of the function
f(x)=sin⁡(x)cos⁡(x)x3f(x) = \frac{\sin(x)\cos(x)}{x^3}
You do not need to simplify your answer.
Differentiation Rules
Find ddx(sin⁡2x+π+cos⁡2x+57)\frac{d}{dx}(\sin^2{x}+\sqrt\pi+\cos^2{x}+5^7)
Finding a derivative: Trigonometric and Exponential Functions
Find the derivative of cos⁡4x+x44x\cos^4x+\frac{x^4}{4^x}.
Finding a derivative: Trigonometric and Exponential Functions
Find the derivative of cos⁡4x+x44x\cos^4x+\frac{x^4}{4^x}.
The Chain Rule: Finding a derivative
Find the derivative of f(x)=cos⁡(e2x)f\left(x\right)=\sqrt{\cos\left(e^{2x}\right)}
Higher Order Derivatives
If f(x)=sin⁡(2x)f\left(x\right)=\sin\left(2x\right), find f(21)(π2)f^{\left(21\right)}\left(\frac{\pi}{2}\right).
(i.e. find the 21st derivative at the point π2\frac{\pi}{2})
Derivatives: Trigonometric Functions
Find the equation of the tangent line to f(x)=(x2−2)sin⁡x+2xcos⁡xf(x) = (x^2 - 2) \sin x+2x \cos x at x=πx = \pi.
Derivatives: Trigonometric Functions
Find f′(π)+f′′(π)f'(\pi)+f''(\pi) if 𝑓(𝑥)=𝑥cos⁡𝑥−sin⁡𝑥𝑓(𝑥) = 𝑥 \cos 𝑥 − \sin 𝑥 .
Derivatives: Trigonometric Functions
Find the derivative of the following function: f(x)=sin⁡xcos⁡x+x3f(x)=\sin x\cos x+\sqrt[3]{x}
Derivatives: Trigonometric Functions
Given that 𝑓(𝑥)=𝑥sin⁡𝑥𝑓(𝑥) = 𝑥\sin 𝑥, find f′ (3π2)f'\ \left(\dfrac{3\pi}{2}\right) .
Derivatives: Trigonometric Functions
Find the derivative of g(x)=sin⁡(πx+1)−cos⁡(πx+32)g\left(x\right)=\sin\left(\pi x+1\right)-\cos\left(\dfrac{\pi x+3}{2}\right)
Derivatives: Trigonometric Functions
Find the derivative of  f(x)=3x2cos⁡(2x)\ f\left(x\right)=3x^2\cos\left(2x\right) .
Derivatives: Exponential and Trigonometric Functions
Differentiate y=csc⁡(1−35x)y=\csc(1-3^{5\sqrt x})
Product and Chain
Q:\textbf{Q:} Find the derivative of h(θ)=θ2−1cos⁡(3θ−1)e−5θh(\theta)=\sqrt{\theta^2-1}\cos(3\theta-1)e^{-5\theta}
Practice: Chain Rule with Given Values
Q.\textbf{Q.} Given that g(1)=g′(1)=1g(1)=g^{\prime}(1)=1, find f′(π/2)f^{\prime}(\pi/2) if f(x)=g(sin⁡x)x+1\displaystyle f(x)=\frac{\sqrt{g(\sin{x})}}{x+1}.
Second Derivative with Trig
Find the second derivative for the function g(s)=1tan⁡s+2es−cot⁡scos⁡s\displaystyle g(s)=\frac{1}{\tan s} +2e^{s}-\frac{\cot s}{\cos s}
Higher Order Trig Derivative
Find y23y^{23} given y=sin⁡x\displaystyle y=\sin x
Quotient with Trig
Find the derivative of h(t)=cos⁡tsin⁡t(1+cos⁡t)\displaystyle h(t)=\frac{\cos t}{\sin t(1+\cos t)}
Tangent Line with Product Rule
Find the equation of the tangent line to the graph of 𝑦=sin⁡𝑥+3𝑥2cos⁡𝑥𝑦 = \sin 𝑥 + 3𝑥^2 \cos 𝑥 at the point where x=π2\displaystyle x=\frac{\pi}{2}
Product and Chain
Q:\textbf{Q:} Find the derivative of h(θ)=θ2−1cos⁡(3θ−1)e−5θh(\theta)=\sqrt{\theta^2-1}\cos(3\theta-1)e^{-5\theta}
Derivatives: Trigonometric and Exponential Functions
Calculate the derivative of the following functions.
f(x)=ex2sec(x)\displaystyle f(x)=e^{x^2\text{sec}(x)}
Find the derivative of the following function.
f(x)=x2+1+cos⁡x\displaystyle f(x)=\sqrt{x^2+1+\cos{x}}
Derivatives: Trigonometric Functions
Find the slope of the tangent line to the following function at x=π/6:x=\pi/6:
g(x)=e−12ln⁡(sin⁡x)g(x)=e^{-\frac{1}{2}\ln(\sin x)}
Derivatives: Trigonometric Functions
Given that g(1)=g′(1)=1g(1)=g'(1)=1, find f′(π/2) if f(x)=g(sin⁡x)x+1f'(\pi/2) \text{ if }f(x)=\frac{\sqrt{g(\sin x)}}{x+1}
Derivatives: Trigonometric Functions
Find
d22dx22(−10cos⁡(x))\frac{d^{22}}{dx^{22}}(-10\cos(x))
Derivatives: Trigonometric Functions
Find the equation of the tangent line and the normal line to f(x)=(x2−2)sin⁡x+2xcos⁡xf(x)=(x^2-2)\sin x+2x\cos x at x=πx=\pi
Derivatives: Trigonometric and Exponential Functions
Compute the derivative of
f(x)=tan⁡(e2x)f(x)=\tan(e^{2x})
Derivatives: Trigonometric Functions
Find the equation of the line tangent to the graph of f(x)=x3sin⁡(πx)f(x)=x^3\sin(\pi x) at the point x=2.x=2.
Practice: Complicated Chain-Rule
Find an expression for the derivative of the following function:
f(x)=cos⁡(e2x)f\left(x\right)=\sqrt{\cos \left(e^{2x}\right)}
Practice: Chain Rule
Find the derivative of the following function:
y=esin⁡(x)y=e^{\sin(\sqrt{x})}
Practice: Equation of Tangent Line
Practice: Equation of Tangent Line
Find the equation of the tangent line to the graph y=sin⁡x+3x2cos⁡xy=\sin x+3x^2\cos x at the point where x=π2x=\frac{\pi}{2}.
Find the equation of the tangent line of the graph y=excos⁡ xy=\frac{e^x}{\cos\ x}at the point where 𝑥 = 0.
Derivatives: Trigonometric Functions
Find f′(π)+f′′(π)f'(\pi)+f''(\pi) if 𝑓(𝑥)=𝑥cos⁡𝑥−sin⁡𝑥𝑓(𝑥) = 𝑥 \cos 𝑥 − \sin 𝑥 .
Derivatives: Trigonometric Functions
Find the derivative of the following function: f(x)=sin⁡xcos⁡x+x3f(x)=\sin x\cos x+\sqrt[3]{x}
Derivatives: Trigonometric Functions
Given that 𝑓(𝑥)=𝑥sin⁡𝑥𝑓(𝑥) = 𝑥\sin 𝑥, find f′ (3π2)f'\ \left(\dfrac{3\pi}{2}\right) .
Derivatives: Trigonometric Functions
Find the derivative of g(x)=sin⁡(πx+1)−cos⁡(πx+32)g\left(x\right)=\sin\left(\pi x+1\right)-\cos\left(\dfrac{\pi x+3}{2}\right)
Derivatives: Trigonometric Functions
Find the equation of the tangent line to f(x)=(x2−2)sin⁡x+2xcos⁡xf(x) = (x^2 - 2) \sin x+2x \cos x at x=πx = \pi.
Practice: Equation of Tangent Line
Practice: Equation of Tangent Line
Find the equation of the tangent line to the graph y=sin⁡x+3x2cos⁡xy=\sin x+3x^2\cos x at the point where x=π2x=\frac{\pi}{2}.
Practice: Equation of Tangent Line
Practice: Equation of Tangent Line
Find the equation of the tangent line to the graph y=sin⁡x+3x2cos⁡xy=\sin x+3x^2\cos x at the point where x=π2x=\frac{\pi}{2}.
Practice: Equation of Tangent Line
Practice: Equation of Tangent Line
Find the equation of the tangent line to the graph y=sin⁡x+3x2cos⁡xy=\sin x+3x^2\cos x at the point where x=π2x=\frac{\pi}{2}.
Derivatives: Trigonometric Functions
Find the derivative of  f(x)=3x2cos⁡(2x)\ f\left(x\right)=3x^2\cos\left(2x\right) .
Derivatives: Exponential and Trigonometric Functions
Differentiate y=csc⁡(1−35x)y=\csc(1-3^{5\sqrt x})
Product and Chain
Q:\textbf{Q:} Find the derivative of h(θ)=θ2−1cos⁡(3θ−1)e−5θh(\theta)=\sqrt{\theta^2-1}\cos(3\theta-1)e^{-5\theta}
Practice: Chain Rule with Given Values
Q.\textbf{Q.} Given that g(1)=g′(1)=1g(1)=g^{\prime}(1)=1, find f′(π/2)f^{\prime}(\pi/2) if f(x)=g(sin⁡x)x+1\displaystyle f(x)=\frac{\sqrt{g(\sin{x})}}{x+1}.
Higher Order Trig Derivative
Find y23y^{23} given y=sin⁡x\displaystyle y=\sin x
Second Derivative with Trig
Find the second derivative for the function g(s)=1tan⁡s+2es−cot⁡scos⁡s\displaystyle g(s)=\frac{1}{\tan s} +2e^{s}-\frac{\cot s}{\cos s}
Quotient with Trig
Find the derivative of h(t)=cos⁡tsin⁡t(1+cos⁡t)\displaystyle h(t)=\frac{\cos t}{\sin t(1+\cos t)}
Tangent Line with Product Rule
Find the equation of the tangent line to the graph of 𝑦=sin⁡𝑥+3𝑥2cos⁡𝑥𝑦 = \sin 𝑥 + 3𝑥^2 \cos 𝑥 at the point where x=π2\displaystyle x=\frac{\pi}{2}
The Chain Rule" Derivatives of Trigonometric Functions
Calculate the derivative of the function f(x)=tan⁡(x3)f\left(x\right)=\sqrt{\tan\left(x^3\right)}
Derivatives of Trigonometric Functions
Determine the point(s), if any, where the function y=cos⁡2x−sin⁡2xy=\cos^2x-\sin^2x has a horizontal tangent line.
Derivatives: Trigonometric and Logarithmic Functions
Find ddy[(y5+ln⁡y)tan⁡y]\displaystyle \frac{\text{d}}{\text{d}y}\left[\left(y^5+\ln y\right)\tan y\right].
Derivatives: Trigonometric and Exponential Functions
Evaluate ddx(exsin⁡x)\displaystyle \frac{\text{d}}{\text{d}x}\left(e^x\sin x\right).
Find the derivative of the function f(x)=sin⁡xcos⁡x+x3f(x)=\sin x\cos x+\sqrt[3]{x}.
Find the equation of the tangent line to f(x)=(x2−2)sin⁡x+2xcos⁡xf(x) = (x^2 - 2) \sin x+2x \cos x at x=πx = \pi.
Find the derivative of 𝑓(𝑥) = 𝑒𝑥 tan 𝑥 + 3.
Given the function f(x)=g(x)h(x)−excos⁡xf\left(x\right)=\frac{g\left(x\right)}{h\left(x\right)}-e^x\cos x, g(π2)=2g\left(\frac{\pi}{2}\right)=2, g′(π2)=3g'\left(\frac{\pi}{2}\right)=3, h(π2)=1h\left(\frac{\pi}{2}\right)=1, and h′(π2)=2h'\left(\frac{\pi}{2}\right)=2, find f′(π2)f'\left(\frac{\pi}{2}\right).
Finding a derivative: Trigonometric and Exponential Functions
Find the derivative of cos⁡4x+x44x\cos^4x+\frac{x^4}{4^x}.
Practice: trig and inverse trig derivatives
Practice: trig and inverse trig derivatives
Find the derivative of the following trigonometric and inverse trigonometric functions:
a) f(x)=arcsin⁡(3x2)f(x)=\arcsin(3x^2)
The Chain Rule: Finding a derivative
Find the derivative of f(x)=cos⁡(e2x)f\left(x\right)=\sqrt{\cos\left(e^{2x}\right)}
Higher Order Derivatives
If f(x)=sin⁡(2x)f\left(x\right)=\sin\left(2x\right), find f(21)(π2)f^{\left(21\right)}\left(\frac{\pi}{2}\right).
(i.e. find the 21st derivative at the point π2\frac{\pi}{2})
Differentiating functions
Find the derivative of the function
f(x)=sin⁡(x)cos⁡(x)x3f(x) = \frac{\sin(x)\cos(x)}{x^3}
You do not need to simplify your answer.
Practice: Chain Rule
Find the derivative of the following function:
y=sin⁡(x)y=\sin(\sqrt{x})
Differentiation Rules
Find ddx(sin⁡2x+π+cos⁡2x+57)\frac{d}{dx}(\sin^2{x}+\sqrt\pi+\cos^2{x}+5^7)
Derivatives: Trigonometric Functions
Find f′(π/2)f^{\prime}(\pi/2)if f(x)=g(sin⁡x)x+1\displaystyle f(x)=\frac{\sqrt{g(\sin{x})}}{x+1}, where g(1)=g′(1)=1. g(1)=g^{\prime}(1)=1.
Derivatives: Trigonometric Functions
Find the derivative of the following function.
f(x)=x2+1+cos⁡x\displaystyle f(x)=\sqrt{x^2+1+\cos{x}}
Practice: The Chain Rule
Find the derivative of the following function
y=esin⁡xy=e^{\sin\sqrt{x}}
Derivatives: Trigonometric and Exponential Functions
Calculate the derivative of the following functions.
f(x)=tan⁡(arccos(e4x))\displaystyle f(x)=\tan(\text{arccos}(e^{4x}))
More Derivatives of Inverse Trig Functions Questions:
Inverse trigonometric derivative
Find the derivative of g(x)=(cos⁡−1x)(sin⁡−1x)g(x)=\left(\cos^{-1}x\right)\left(\sin^{-1}x\right).
Derivatives: Inverse Trigonometric Functions
Given that f(x)=2−arcsin⁡(x)f\left(x\right)=2-\arcsin\left(x\right), find f′(0)f'\left(0\right)
Inverse Trigonometric Functions
Compute the derivative of f(x)=arcsin⁡(x+2)f\left(x\right)=\arcsin\left(x+2\right)
Derivatives: Inverse Trigonometric Functions
Given that f(x)=(2−arcsin⁡(x2))3f\left(x\right)=\left(2-\arcsin\left(x^2\right)\right)^3, find f′(0)f'\left(0\right)
The Chain Rule
Q:\bf{Q:} Find (33x−x4+2cos⁡−1(x)−2025)′(3^{3x-x^{4}}+2^{\cos^{-1}(x)}-2025)'
Derivatives
Find the derivative of the following functions at the given values
a) f(x)=arcsin⁡(x2)x3−1f\left(x\right)=\frac{\arcsin\left(x^2\right)}{x^3-1} at x=0x=0
b) g(x)=(cos⁡x)exg\left(x\right)=\left(\cos x\right)^{e^x} at x=0x=0
Finding a Derivative
If f(x)=tan⁡−1(3sin⁡x)f(x)=\tan^{-1}\left(3^{\sin x}\right), find f′(π)f'(\pi).
The Chain Rule
Q:\bf{Q:} Find (33x−x4+2cos−1(x)−2019)′(3^{3x-x^4}+2^{cos^{-1}(x)}-2019)'
Inverse trigonometric derivative
Find the derivative of g(x)=cos⁡−1(sin⁡−1x)g(x)=\cos^{-1}\left(\sin^{-1}x\right).
Find ddx(arctan⁡(etan⁡x))\frac{d}{dx}\left(\arctan\left(e^{\tan x}\right)\right).
Inverse trigonometric derivative
Find the derivative of g(x)=cos⁡−1(sin⁡−1x)g(x)=\cos^{-1}\left(\sin^{-1}x\right).
Inverse Trigonometric Functions
Compute the derivative of f(x)=arcsin(x+2x−1)\displaystyle f(x)=\text{arcsin}\left(\frac{x+2}{x-1}\right)
Derivatives: Inverse Trigonometric Functions
Given that f(x)=(2−arcsin⁡(x2))3f\left(x\right)=\left(2-\arcsin\left(x^2\right)\right)^3, find f′(0)f'\left(0\right).
Evaluating a derivative at a point
Evaluate (cos⁡−1(x2))′∣0\left( \cos^{-1}\left(x^2\right) \right)'\rvert_{0}
Inverse Trig Derivative
Find the equation of the line tangent to
f(x)=sin⁡−1x1+xf(x)=\frac{\sin^{-1}x}{1+x}
at x=1/2x=1/2.
Inverse Trigonometric Functions
Compute the derivative of f(x)=arcsin(x+2x−1)\displaystyle f(x)=\text{arcsin}\left(\frac{x+2}{x-1}\right)
Derivatives: Inverse Trigonometric Functions
Practice: Finding a Derivative
Given that f(x)=(2−arcsin⁡(x2))3f\left(x\right)=\left(2-\arcsin\left(x^2\right)\right)^3, find f′(0)f'\left(0\right).
Evaluating a derivative at a point
Evaluate (cos⁡−1(x2))′∣0\left( \cos^{-1}\left(x^2\right) \right)'\rvert_{0}
Inverse Trig Derivative
Find the equation of the line tangent to
f(x)=sin⁡−1x1+xf(x)=\frac{\sin^{-1}x}{1+x}
at x=1/2x=1/2.
Inverse trigonometric derivative
Find the derivative of g(x)=cos⁡−1(sin⁡−1x)g(x)=\cos^{-1}\left(\sin^{-1}x\right).
Practice: Chain Rule
Find the derivative of y=ln⁡(arctan⁡x)y=\ln(\arctan x)
Find ddx(arctan⁡(etan⁡x))\frac{d}{dx}\left(\arctan\left(e^{\tan x}\right)\right).
Evaluating a derivative at a point
Evaluate (cos⁡−1(x2))′∣0\left( \cos^{-1}\left(x^2\right) \right)'\rvert_{0}
The Chain Rule
Q:\bf{Q:} Find (33x−x4+2cos−1(x)−2019)′(3^{3x-x^4}+2^{cos^{-1}(x)}-2019)'
Derivatives: Inverse Trigonometric Functions
Find [tan⁡−1(x3)]′[\tan^{-1}(\sqrt[3]x)]'
Derivatives: Inverse Trigonometric Functions
Show thatddxsin⁡−1x=11−x2\dfrac{d}{dx}\sin^{-1}x=\dfrac{1}{\sqrt{1-x^2}}
Practice: Derivative of Inverse Trig
Q:\textbf{Q:} Show that ddxcos⁡−1x=−11−x2.\displaystyle\frac{d}{dx}\cos^{-1}x=-\frac{1}{\sqrt{1-x^2}}.
Practice: Tangent Line with Inverse Trig
Q:\textbf{Q:} Find the equation of the tangent line to the graph of f(x)=cot⁡−1(x2)\displaystyle f(x)=\cot^{-1}(x^{2}) at the point x=2x=2.
Inverse Trig with Log
Compute the derivative of f(x)=arctan(log10x)\displaystyle f(x)=\text{arctan}\left(\text{log}_{10}x\right)
Inverse Trigonometric Functions
Compute the derivative of f(x)=arcsin(x+2x−1)\displaystyle f(x)=\text{arcsin}\left(\frac{x+2}{x-1}\right)
Derivatives
Find the derivative of the following functions at the given values
a) f(x)=arcsin⁡(x2)x3−1f\left(x\right)=\frac{\arcsin\left(x^2\right)}{x^3-1} at x=0x=0
b) g(x)=(cos⁡x)exg\left(x\right)=\left(\cos x\right)^{e^x} at x=0x=0
Derivatives: Exponential and Trigonometric Functions
Find the derivative of
f(x)=5x6−sec⁡−1(e2x)+cos⁡−1xf(x)=5x^6-\sec^{-1}(e^{2x})+\cos^{-1}x
Inverse Trig Derivative
Find the equation of the line tangent to
f(x)=sin⁡−1x1+xf(x)=\frac{\sin^{-1}x}{1+x}
at x=1/2x=1/2.
Derivatives: Inverse Trigonometric Functions
Find f′(x)f'(x) if f(x)=arctan⁡xf(x)=\arctan \sqrt{x}
Differentiate the following
y=tan⁡−1x2−1y=\tan^{-1}\sqrt{x^2-1}
Find the derivative of the following function.
f(x)=arcsinx\displaystyle f(x)=\sqrt{\text{arcsin{x}}}
The Chain Rule
Q:\bf{Q:} Find (33x−x4+2cos−1(x)−2019)′(3^{3x-x^4}+2^{cos^{-1}(x)}-2019)'
Derivatives: Inverse Trigonometric Functions
Find [tan⁡−1(x3)]′[\tan^{-1}(\sqrt[3]x)]'
Derivatives: Inverse Trigonometric Functions
Show thatddxsin⁡−1x=11−x2\dfrac{d}{dx}\sin^{-1}x=\dfrac{1}{\sqrt{1-x^2}}
Practice: Derivative of Inverse Trig
Q:\textbf{Q:} Show that ddxcos⁡−1x=−11−x2.\displaystyle\frac{d}{dx}\cos^{-1}x=-\frac{1}{\sqrt{1-x^2}}.
Practice: Tangent Line with Inverse Trig
Q:\textbf{Q:} Find the equation of the tangent line to the graph of f(x)=cot⁡−1(x2)\displaystyle f(x)=\cot^{-1}(x^{2}) at the point x=2x=2.
Inverse Trig with Log
Compute the derivative of f(x)=arctan(log10x)\displaystyle f(x)=\text{arctan}\left(\text{log}_{10}x\right)
Inverse Trigonometric Functions
Compute the derivative of f(x)=arcsin(x+2x−1)\displaystyle f(x)=\text{arcsin}\left(\frac{x+2}{x-1}\right)
Derivatives: Exponential Functions, Inverse Trigonometric Functions
Compute the derivative of f(x)=sec⁡(32x)f(x)=\sec(3^{2x}) .
Derivatives: Logarithmic and Inverse Trigonometric Functions
Evaluate ddx(sin⁡−1xsin⁡xlog⁡3x)\displaystyle \frac{\text{d}}{\text{d}x}\left( \frac{\sin^{-1}x\sin x}{\log_3x}\right).
Find the derivative of f(x)=arcsin⁡(ex+1)f(x)=\arcsin(e^{x+1}).
Inverse trigonometric derivative
Find the derivative of g(x)=cos⁡−1(sin⁡−1x)g(x)=\cos^{-1}\left(\sin^{-1}x\right).
Derivatives: Inverse Trigonometric Functions
Find the derivative of f(x)=(arccos⁡x)2sec⁡xf\left(x\right)=\frac{\left(\arccos x\right)^2}{\sec x}.
Evaluating a derivative at a point
Evaluate (cos⁡−1(x2))′∣0\left( \cos^{-1}\left(x^2\right) \right)'\rvert_{0}
Practice: Chain Rule
Find the derivative of y=ln⁡(arctan⁡x)y=\ln(\arctan x)
Find ddx(arctan⁡(etan⁡x))\frac{d}{dx}\left(\arctan\left(e^{\tan x}\right)\right).
Practice: trig and inverse trig derivatives
Practice: trig and inverse trig derivatives
Find the derivative of the following trigonometric and inverse trigonometric functions:
a) f(x)=arcsin⁡(3x2)f(x)=\arcsin(3x^2)
Derivatives: Inverse Trigonometric Functions
Practice: Finding a Derivative
Given that f(x)=(2−arcsin⁡(x2))3f\left(x\right)=\left(2-\arcsin\left(x^2\right)\right)^3, find f′(0)f'\left(0\right).
Finding a Derivative
Practice: Finding a Derivative
(tan⁡−1(3sin⁡x))′∣x=π=\left(\tan^{-1}\left(3^{\sin x}\right)\right)'|_{x=\pi}=
Linear Approximation and Trigonometric Derivative
a) Show that for f(x)=arcsin⁡(x)f(x) = \arcsin(x) , that f′(x)=11−x2f'(x) = \frac{1}{\sqrt{1 - x^2}} .
b) Use linear approximation at a suitable close value to estimate arcsin⁡(0.1)\arcsin(0.1) . Your solution may be left in terms of fractions.
Derivatives: Inverse Trigonometric Functions
Find the derivative of the following function.
f(x)=1arcsin(x)+arcsin(1x)\displaystyle f(x)=\frac{1}{\text{arcsin}(x)}+\text{arcsin}\left(\frac{1}{x}\right)
Inverse Trigonometric Functions
Calculate the derivative of the following functions.
f(x)=arcsin(ex+1)\displaystyle f(x)=\text{arcsin}(e^{x+1})
Derivatives: Trigonometric and Exponential Functions
Calculate the derivative of the following functions.
f(x)=tan⁡(arccos(e4x))\displaystyle f(x)=\tan(\text{arccos}(e^{4x}))
More The Chain Rule Questions:
The Chain Rule
Q:\bf{Q:} Find (33x−x4+2cos⁡−1(x)−2025)′(3^{3x-x^{4}}+2^{\cos^{-1}(x)}-2025)'
Derivatives
Find the derivative of the following functions at the given values
a) f(x)=arcsin⁡(x2)x3−1f\left(x\right)=\frac{\arcsin\left(x^2\right)}{x^3-1} at x=0x=0
b) g(x)=(cos⁡x)exg\left(x\right)=\left(\cos x\right)^{e^x} at x=0x=0
Finding a Derivative
If f(x)=tan⁡−1(3sin⁡x)f(x)=\tan^{-1}\left(3^{\sin x}\right), find f′(π)f'(\pi).
The Chain Rule
Q:\bf{Q:} Find (33x−x4+2cos−1(x)−2019)′(3^{3x-x^4}+2^{cos^{-1}(x)}-2019)'
The Chain Rule
Find the derivative of h(x)=f(g(x))h(x)=f(g(x)) if f(x)=x2+1 f(x)=x^2+1 and g(x)=3xg(x)=3\sqrt{x} .
The Chain Rule
Find the derivative of h(x)=f(g(x))h(x)=f(g(x)) if f(x)=x2+1 f(x)=x^2+1 and g(x)=3xg(x)=3\sqrt{x} .
The Chain Rule
Find g′(0)g'(0) for g(x)=f(x2)+f(x),g(x)=\sqrt{f(x^2)} + f(x), given f(0)=2 and f′(0)=1.f(0)=2 \text{ and }f'(0)=1.
The Chain Rule
If 2−g(x)=x2+2[f(x)]2−x3g(x)2-g\left(x\right)=x^2+2\left[f\left(x\right)\right]^2-x^3g\left(x\right), f(0)=1 , f′(0)=−2, g(0)=−1 and  g′′(0)=3f\left(0\right)=1\ ,\ f'\left(0\right)=-2,\ g\left(0\right)=-1\ \text{and}\ \ g''\left(0\right)=3, then the value of f′′(0)f''\left(0\right) is equal to
Find g′(0)g^{\prime}(0) for g(x)=f(x2)+f(x)g(x)=\sqrt{f(x^2)}+f(x), given f(0)=2f(0)=2 and f′(0)=1f^{\prime}(0)=1.
The Chain Rule
Find the derivative of the following function.
f(x)=(2x+1x2+1)3\displaystyle f(x)=\left(\frac{2x+1}{x^2+1}\right)^3
Find ddx(arctan⁡(etan⁡x))\frac{d}{dx}\left(\arctan\left(e^{\tan x}\right)\right).
The Chain Rule
The derivative of y=3exy=3^{e^x} is
The Chain Rule
The derivative of y=3exy=3^{e^x} is
The Chain Rule
The derivative of y=3exy=3^{e^x} is
The Chain Rule
The derivative of y=3exy=3^{e^x} is
Practice: Chain Rule
Find the derivative of y=ln⁡(arctan⁡x)y=\ln(\arctan x)
Find ddx(arctan⁡(etan⁡x))\frac{d}{dx}\left(\arctan\left(e^{\tan x}\right)\right).
The Chain Rule: Finding a derivative
Find the derivative of f(x)=cos⁡(e2x)f\left(x\right)=\sqrt{\cos\left(e^{2x}\right)}
The Chain Rule
Given this table of values for f, g, f′, and g′f,\ g,\ f',\ \text{and}\ g' below, answer the following questions.
xf(x)f′(x)f′′(x)g(x)g′(x)00−1−523π2π10452−2−410π2−3\begin{array}{|c|c|c|c|c|c|} \hline x&f(x)&f'(x)&f''(x)&g(x)&g'(x)\\ \hline 0&0&-1&-5&2&3\\ \hline \frac{\pi}{2}&\pi&1&0&4&5\\ \hline 2&-2&-4&10&\frac{\pi}{2}&-3\\ \hline \end{array}
The Chain Rule
Q:\bf{Q:} Find (33x−x4+2cos−1(x)−2019)′(3^{3x-x^4}+2^{cos^{-1}(x)}-2019)'
The Chain Rule
Find the derivative of h(x)=f(g(x))h(x)=f(g(x)) if f(x)=x2+1 f(x)=x^2+1 and g(x)=3xg(x)=3\sqrt{x} .
Practice: Chain Rule with Given Values
Q.\textbf{Q.} Given that g(1)=g′(1)=1g(1)=g^{\prime}(1)=1, find f′(π/2)f^{\prime}(\pi/2) if f(x)=g(sin⁡x)x+1\displaystyle f(x)=\frac{\sqrt{g(\sin{x})}}{x+1}.
Practice: Chain Rule with Given Values
Q:\textbf{Q:} Given that f(1)=5, g(1)=−2, f′(1)=2f(1)=5,\ g(1)=-2,\ f'(1)=2 and g′(1)=−1/2g'(1)=-1/2, find the derivative of f(x)−g2(x)\sqrt{f(x)-g^2(x)} at x=1x=1.
The Quotient and Chain Rules
Let f(x)=x2−6x−3\displaystyle f(x) = \frac{\sqrt{x^2 - 6}}{x - 3}.
The chain rule
Find the derivative of h(x)=log⁡(cos⁡(x))h(x) = \log(\cos(x)). Remember that log⁡x=log⁡ex=ln⁡x\log x = \log_e x = \ln x.
Derivatives
Find the derivative of the following functions at the given values
a) f(x)=arcsin⁡(x2)x3−1f\left(x\right)=\frac{\arcsin\left(x^2\right)}{x^3-1} at x=0x=0
b) g(x)=(cos⁡x)exg\left(x\right)=\left(\cos x\right)^{e^x} at x=0x=0
The chain rule: Logarithmic Derivatives
The derivative of log⁡3(e3x)\log_3\left(e^{3x}\right) is
The chain rule
The derivative of y=(2−x7)500y=\left(2-x^7\right)^{500} is
The Chain Rule
If 2−g(x)=x2+2[f(x)]2−x3g(x)2-g\left(x\right)=x^2+2\left[f\left(x\right)\right]^2-x^3g\left(x\right), f(0)=1 , f′(0)=−2, g(0)=−1 and  g′′(0)=3f\left(0\right)=1\ ,\ f'\left(0\right)=-2,\ g\left(0\right)=-1\ \text{and}\ \ g''\left(0\right)=3, then the value of f′′(0)f''\left(0\right) is equal to
Find the derivative of h(x)=f(g(x))   if  f(x)=x2+1h(x)=f(g(x))\,\, \text{ if }\, f(x)=x^2+1 and g(x)=3xg(x)=3\sqrt{x}.
Find g′(0)g^{\prime}(0) for g(x)=f(x2)+f(x)g(x)=\sqrt{f(x^2)}+f(x), given f(0)=2f(0)=2and f′(0)=1f^{\prime}(0)=1.
Find the derivative of h(x)=f(g(x))   if  f(x)=x2+1h(x)=f(g(x))\,\, \text{ if }\, f(x)=x^2+1 and g(x)=3xg(x)=3\sqrt{x}.
Find the derivative of the following function.
f(x)=(2x+1x2+1)3\displaystyle f(x)=\left(\frac{2x+1}{x^2+1}\right)^3
The Chain Rule
Find g′(0)g'(0) for g(x)=f(x2)+f(x),g(x)=\sqrt{f(x^2)} + f(x), given f(0)=2 and f′(0)=1.f(0)=2 \text{ and }f'(0)=1.
The Chain Rule
Compute the derivative of
f(x)=x4+4x4+4f(x) = \sqrt{x^4 + \frac{4}{x^4}+4}
Practice: Chain Rule
Find the derivative of the following function:
y=sin⁡(x)y=\sin(\sqrt{x})
Practice: Complicated Chain-Rule
Find an expression for the derivative of the following function:
f(x)=cos⁡(e2x)f\left(x\right)=\sqrt{\cos \left(e^{2x}\right)}
Practice: Chain Rule
Find the derivative of the following function:
y=esin⁡(x)y=e^{\sin(\sqrt{x})}
Find the derivative of the following function.
f(x)=arcsinx\displaystyle f(x)=\sqrt{\text{arcsin{x}}}
The Chain Rule
Q:\bf{Q:} Find (33x−x4+2cos−1(x)−2019)′(3^{3x-x^4}+2^{cos^{-1}(x)}-2019)'
The Chain Rule
Find the derivative of h(x)=f(g(x))h(x)=f(g(x)) if f(x)=x2+1 f(x)=x^2+1 and g(x)=3xg(x)=3\sqrt{x} .
Practice: Chain Rule with Given Values
Q.\textbf{Q.} Given that g(1)=g′(1)=1g(1)=g^{\prime}(1)=1, find f′(π/2)f^{\prime}(\pi/2) if f(x)=g(sin⁡x)x+1\displaystyle f(x)=\frac{\sqrt{g(\sin{x})}}{x+1}.
Practice: Chain Rule with Given Values
Q:\textbf{Q:} Given that f(1)=5, g(1)=−2, f′(1)=2f(1)=5,\ g(1)=-2,\ f'(1)=2 and g′(1)=−1/2g'(1)=-1/2, find the derivative of f(x)−g2(x)\sqrt{f(x)-g^2(x)} at x=1x=1.
Given the information g(x)=1+[f(x)]2  such that f(1)=2,f′(1)=−3g(x)=\sqrt{1+[f(x)]^2} \ \text{ such that } f(1)=2,f'(1)=-3. Find g′(1).g'(1). Express your answer as a fraction in lowest terms. If the answer is negative, put the negative sign in front of the entire fraction.
The Chain Rule" Derivatives of Trigonometric Functions
Calculate the derivative of the function f(x)=tan⁡(x3)f\left(x\right)=\sqrt{\tan\left(x^3\right)}
The Chain Rule
If g(t)=9+f(t)3g\left(t\right)=\sqrt[3]{9+f(t)} , write an expression for the derivative g′(t)g'\left(t\right)
The Chain Rule
Find the derivative of the following function.
f(x)=(2x+1x2+1)3\displaystyle f(x)=\left(\frac{2x+1}{x^2+1}\right)^3
Practice: Chain Rule
Find the derivative of y=ln⁡(arctan⁡x)y=\ln(\arctan x)
Find ddx(arctan⁡(etan⁡x))\frac{d}{dx}\left(\arctan\left(e^{\tan x}\right)\right).
Practice: trig and inverse trig derivatives
Practice: trig and inverse trig derivatives
Find the derivative of the following trigonometric and inverse trigonometric functions:
a) f(x)=arcsin⁡(3x2)f(x)=\arcsin(3x^2)
The Chain Rule: Finding a derivative
Find the derivative of f(x)=cos⁡(e2x)f\left(x\right)=\sqrt{\cos\left(e^{2x}\right)}
(Duplicated)
Find f′(1)f'\left(1\right) if f(x)=x2 g(x)f\left(x\right)=\sqrt{x^2\ g\left(x\right)}, g(1)=1g\left(1\right)=1 and g′(1)=2g'\left(1\right)=2.
(Duplicated)
If f(x)=(g(x)−h(x))5f\left(x\right)=\left(g\left(x\right)-h\left(x\right)\right)^5, g(0)=2, h(0)=0, g′(0)=6g\left(0\right)=2,\ h\left(0\right)=0,\ g'\left(0\right)=6, and h′(0)=−4h'\left(0\right)=-4, find f′(0)f'\left(0\right).
The Chain Rule
The derivative of y=3exy=3^{e^x} is
The Chain Rule
Given this table of values for f, g, f′, and g′f,\ g,\ f',\ \text{and}\ g' below, answer the following questions.
xf(x)f′(x)f′′(x)g(x)g′(x)00−1−523π2π10452−2−410π2−3\begin{array}{|c|c|c|c|c|c|} \hline x&f(x)&f'(x)&f''(x)&g(x)&g'(x)\\ \hline 0&0&-1&-5&2&3\\ \hline \frac{\pi}{2}&\pi&1&0&4&5\\ \hline 2&-2&-4&10&\frac{\pi}{2}&-3\\ \hline \end{array}
Finding a Derivative
Practice: Finding a Derivative
(tan⁡−1(3sin⁡x))′∣x=π=\left(\tan^{-1}\left(3^{\sin x}\right)\right)'|_{x=\pi}=
Practice: The Chain Rule
Find the derivative of the following function
y=esin⁡xy=e^{\sin\sqrt{x}}
Derivatives: Trigonometric and Exponential Functions
Calculate the derivative of the following functions.
f(x)=tan⁡(arccos(e4x))\displaystyle f(x)=\tan(\text{arccos}(e^{4x}))