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# 2.4: Derivatives of Trigonometric Functions

Difficulty Level: At Grade Created by: CK-12

## Learning Objectives

A student will be able to:

• Compute the derivatives of various trigonometric functions.

If the angle h\begin{align*}h\end{align*} is measured in radians,

limh0sinhh=1\begin{align*} \lim_{h \to 0} \frac{\sin h} {h} = 1\end{align*} and limh01coshh=0.\begin{align*}\lim_{h \to 0} \frac{1 - \cos h} {h} = 0.\end{align*}

We can use these limits to find an expression for the derivative of the six trigonometric functions sinx,cosx,tanx,secx,cscx,\begin{align*}\sin x, \cos x, \tan x, \sec x, \csc x,\end{align*} and cotx\begin{align*}\cot x\end{align*}. We first consider the problem of differentiating sinx\begin{align*}\sin x\end{align*}, using the definition of the derivative.

ddx[sinx]=limh0sin(x+h)sinxh

Since

sin(α+β)=sinα cosβ+cosα sinβ.

The derivative becomes

ddx[sinx]=limh0sinxcosh+cosxsinhsinxh=limh0[sinx(cosh1h)+cosx(sinhh)]=sinxlimh0(1coshh)+cosxlimh0(sinhh)=sinx(0)+cosx(1)=cosx.

Therefore,

It will be left as an exercise to prove that

The derivatives of the remaining trigonometric functions are shown in the table below.

Derivatives of Trigonometric Functions

Keep in mind that for all the derivative formulas for the trigonometric functions, the argument \begin{align*}x\end{align*} is measured in radians.

Example 1:

Show that \begin{align*} \frac {d}{dx} [\tan x] = \sec^2 x.\end{align*}

Solution:

It is possible to prove this relation by the definition of the derivative. However, we use a simpler method.

Since

then

Example 2:

Find \begin{align*}f'(x) \mathrm{~if~} f(x) = x^{2} \cos x + \sin x\end{align*}.

Solution:

Using the product rule and the formulas above, we obtain

Example 3:

Find \begin{align*}dy/dx\end{align*} if \begin{align*} y = \frac {\cos x}{1-\tan x}\end{align*} . What is the slope of the tangent line at \begin{align*}x = \pi/3\end{align*}?

Solution:

Using the quotient rule and the formulas above, we obtain

To calculate the slope of the tangent line, we simply substitute \begin{align*}x = \pi/3\end{align*}:

We finally get the slope to be approximately

Example 4:

If \begin{align*}y = \sec x\end{align*}, find \begin{align*}y'' (\pi/3)\end{align*}.

Solution:

Substituting for \begin{align*}x = \pi/3\end{align*},

Thus \begin{align*}y'' (\pi/3) = 14\end{align*}.

For examples of finding the derivatives of trigonometric functions (4.4), see Math Video Tutorials by James Sousa, The Derivative of Sine and Cosine (9:21).

## Review Questions

Find the derivative \begin{align*}y'\end{align*} of the following functions:

1. \begin{align*}y = x \sin x + 2\end{align*}
2. \begin{align*}y = x^2 \cos x - x \tan x - 1\end{align*}
3. \begin{align*}y = \sin^2 x\end{align*}
4. \begin{align*} y = \frac {\sin x-1}{\sin x+1}\end{align*}
5. \begin{align*} y = \frac {\cos x+\sin x} {\cos x - \sin x}\end{align*}
6. \begin{align*} y = \frac {\sqrt {x}}{\tan x} + 2\end{align*}
7. \begin{align*}y = \csc x \sin x + x\end{align*}
8. \begin{align*} y = \frac {\sec x} {\csc x}\end{align*}
9. If \begin{align*}y = \csc x\end{align*}, find \begin{align*}y'' (\pi/6).\end{align*}
10. Use the definition of the derivative to prove that \begin{align*} \frac {d}{dx}[\cos x] = - \sin x.\end{align*}

## Date Created:

Feb 23, 2012

Oct 30, 2015
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