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Differentiation Rules

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Wize University Calculus 2 Textbook > Review: Derivatives

Derivative Rule

9 Activities

Wize University Calculus 1 Textbook > Derivatives

Derivatives of Trig Functions

6 Activities

Find ddx(sin⁡2x+π+cos⁡2x+57)\frac{d}{dx}(\sin^2{x}+\sqrt\pi+\cos^2{x}+5^7)



More Derivative Rule Questions:
Practice: Derivative Rules (Level 2)
Find the rate of the change of the function f(x)=5x−x32+13x\displaystyle f(x)=\frac{5}{\sqrt{x}}-\frac{\sqrt[3]{x}}{2}+\frac{1}{3x} at the point x=1x=1.
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)
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: Product Rule*
Suppose g(x)g\left(x\right) is differentiable, f(x)=x2g(x)f\left(x\right)=x^2g\left(x\right). Given that f′(3)=30f'\left(3\right)=30, f′′(3)=19f''\left(3\right)=19, g(3)=2g\left(3\right)=2, what is g′(3)g'\left(3\right) and g′′(3)g''\left(3\right)?
Practice: Quotient Rule
Find the derivative of f(x)=x2+3x4/3+1x2+1\displaystyle f(x)=\frac{x^2+3x^{4/3}+1}{x^2+1}
Practice: Derivative Rule (Level 3)
Practice: Derivative Rule
Given f(t)=2t3(t+1t)t\displaystyle f(t)=\frac{2t^3\left(t+\frac{1}{t}\right)}{\sqrt{t}}, find f′(4)f'\left(4\right).
Practice: Multiple Chain Rules
Practice: Multiple Chain Rules
Find the derivative of f(x)=x+x4+x3+x2+1f\left(x\right)=x+\sqrt{x^4+\sqrt{x^3+\sqrt{x^2+1}}} at x=0x=0
Practice: Derivative Rules (Level 2)
Practice: Derivative Rules
Find the rate of the change of the function f(x)=5x−x32+13x\displaystyle f(x)=\frac{5}{\sqrt{x}}-\frac{\sqrt[3]{x}}{2}+\frac{1}{3x} at the point x=1x=1.
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)}
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)
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})
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}))