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Wize University Calculus 1 Textbook > Derivatives

Derivatives of Trig Functions

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Derivatives of Trig Functions

We may often want to take derivatives involving the trigonometric functions. Rather than deriving them each time, it's much easier to simply remember the following derivatives.

Trig Derivatives

The derivatives of the six trigonometric functions are:

(sin⁡x)′=cos⁡x          (cos⁡x)′=−sin⁡x(tan⁡x)′=sec⁡2x      (sec⁡x)′=sec⁡xtan⁡x(cot⁡x)′=−csc⁡2x(csc⁡x)′=−csc⁡xcot⁡x\begin{array}{c} \boxed{(\sin x)'=\cos x}\ \ \ \ \ \hspace{1 cm}\ \ \ \ \ \boxed{(\cos x)'=-\sin x } \\ \\ \boxed{(\tan x)'=\sec^{2} x} \ \ \ \ \ \ \qquad\boxed{(\sec x)'=\sec x\tan x} \\ \\ \boxed{ (\cot x)'=-\csc^{2}x}\qquad\boxed{(\csc x)'=-\csc x\cot x } \end{array}


Wize Tip
Notice how all the derivatives of "CO Functions" have a minus sign.

Wize Concept
The Product, Quotient, and Chain Rules still apply for Trig Derivatives!

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Example: Trig Derivatives


Find ddx(x2sin⁡x)\displaystyle\frac{d}{dx}(x^2\sin{x}).

ddx(x2sin⁡x)\displaystyle \frac{d}{dx}(x^2\sin{x})
=(ddxx2)×sin⁡x+x2×(ddxsin⁡x)\displaystyle =(\frac{d}{dx}x^2)\times\sin{x}+x^2\times(\frac{d}{dx}\sin{x})
=2xsin⁡x+x2cos⁡x\displaystyle =2x\sin{x}+x^2\cos{x}
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Example: Trig Derivatives

Find the derivative of f(x)=1+cos⁡xx−sin⁡x\displaystyle f(x)=\frac{1+\cos x}{x-\sin x}


f(x)=y=1+cos⁡xx−sin⁡x\displaystyle f(x)=y=\frac{1+\cos x}{x-\sin x}

Quotient rule:

y′=−sin⁡x(x−sin⁡x)−(1+cos⁡x)(1−cos⁡x)(x−sin⁡x)2y'=\dfrac{-\sin x(x-\sin x)-(1+\cos x)(1-\cos x)}{(x-\sin x)^2}

=−xsin⁡x+sin⁡2x−1+cos⁡2x(x−sin⁡x)2=\dfrac{-x\sin x+\sin ^2x-1+\cos ^2x}{(x-\sin x)^2}

=−xsin⁡x(x−sin⁡x)2=-\dfrac{x\sin x}{(x-\sin x)^2}
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Example: Trig Derivatives

Find the derivative of the following function

y=sin⁡xy=\sin\sqrt{x}

The outside function is f(x)=sin⁡(x)f(x)=\sin (x) and the inside function is g(x)=x=x−12g(x)=\sqrt{x}=x^{-\frac{1}{2}}.

The chain rule tells us that the derivative is the derivative of the outside function (keeping the inside the same), times the derivative of the inside function:
y′=f′(g(x))×g′(x)y'=f'\left(g\left(x\right)\right)\times g'\left(x\right)
y′=cos⁡(x)×12x−12\displaystyle y'=\cos\left(\sqrt{x}\right)\times\frac{1}{2}x^{-\frac{1}{2}}

Simplifying:
y′=cos⁡(x)2x\displaystyle y'=\frac{\cos\left(\sqrt{x}\right)}{2\sqrt{x}}
Find the derivative of

f(θ)=(sin⁡θ−cos⁡θ)sec⁡θ\displaystyle f(\theta)=(\sin\theta-\cos\theta)\sec\theta



Find y(23)y^{(23)} , the 23rd derivatives, given y=sin⁡x\displaystyle y=\sin x.

Extra Practice
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})
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}))