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Calculus Volume 1

3.5 Derivatives of Trigonometric Functions

Calculus Volume 13.5 Derivatives of Trigonometric Functions

Learning Objectives

  • 3.5.1 Find the derivatives of the sine and cosine function.
  • 3.5.2 Find the derivatives of the standard trigonometric functions.
  • 3.5.3 Calculate the higher-order derivatives of the sine and cosine.

One of the most important types of motion in physics is simple harmonic motion, which is associated with such systems as an object with mass oscillating on a spring. Simple harmonic motion can be described by using either sine or cosine functions. In this section we expand our knowledge of derivative formulas to include derivatives of these and other trigonometric functions. We begin with the derivatives of the sine and cosine functions and then use them to obtain formulas for the derivatives of the remaining four trigonometric functions. Being able to calculate the derivatives of the sine and cosine functions will enable us to find the velocity and acceleration of simple harmonic motion.

Derivatives of the Sine and Cosine Functions

We begin our exploration of the derivative for the sine function by using the formula to make a reasonable guess at its derivative. Recall that for a function f(x),f(x),

f(x)=limh0f(x+h)f(x)h.f(x)=limh0f(x+h)f(x)h.

Consequently, for values of hh very close to 0, f(x)f(x+h)f(x)h.f(x)f(x+h)f(x)h. We see that by using h=0.01,h=0.01,

ddx(sinx)sin(x+0.01)sinx0.01ddx(sinx)sin(x+0.01)sinx0.01

By setting D(x)=sin(x+0.01)sinx0.01D(x)=sin(x+0.01)sinx0.01 and using a graphing utility, we can get a graph of an approximation to the derivative of sinxsinx (Figure 3.25).

The function D(x) = (sin(x + 0.01) − sin x)/0.01 is graphed. It looks a lot like a cosine curve.
Figure 3.25 The graph of the function D(x)D(x) looks a lot like a cosine curve.

Upon inspection, the graph of D(x)D(x) appears to be very close to the graph of the cosine function. Indeed, we will show that

ddx(sinx)=cosx.ddx(sinx)=cosx.

If we were to follow the same steps to approximate the derivative of the cosine function, we would find that

ddx(cosx)=sinx.ddx(cosx)=sinx.

Theorem 3.8

The Derivatives of sin x and cos x

The derivative of the sine function is the cosine and the derivative of the cosine function is the negative sine.

ddx(sinx)=cosxddx(sinx)=cosx
(3.11)
ddx(cosx)=sinxddx(cosx)=sinx
(3.12)

Proof

Because the proofs for ddx(sinx)=cosxddx(sinx)=cosx and ddx(cosx)=sinxddx(cosx)=sinx use similar techniques, we provide only the proof for ddx(sinx)=cosx.ddx(sinx)=cosx. Before beginning, recall two important trigonometric limits we learned in Introduction to Limits:

limh0sinhh=1andlimh0cosh1h=0.limh0sinhh=1andlimh0cosh1h=0.

The graphs of y=(sinh)hy=(sinh)h and y=(cosh1)hy=(cosh1)h are shown in Figure 3.26.

The function y = (sin h)/h and y = (cos h – 1)/h are graphed. They both have discontinuities on the y-axis.
Figure 3.26 These graphs show two important limits needed to establish the derivative formulas for the sine and cosine functions.

We also recall the following trigonometric identity for the sine of the sum of two angles:

sin(x+h)=sinxcosh+cosxsinh.sin(x+h)=sinxcosh+cosxsinh.

Now that we have gathered all the necessary equations and identities, we proceed with the proof.

ddxsinx=limh0sin(x+h)sinxhApply the definitionof the derivative.=limh0sinxcosh+cosxsinhsinxhUse trig identity for the sine of the sum of two angles.=limh0(sinxcoshsinxh+cosxsinhh)Regroup.=limh0(sinx(cosh1h)+cosx(sinhh))Factor outsinxandcosx. =sinx·0+cosx·1Apply trig limit formulas.=cosxSimplify.ddxsinx=limh0sin(x+h)sinxhApply the definitionof the derivative.=limh0sinxcosh+cosxsinhsinxhUse trig identity for the sine of the sum of two angles.=limh0(sinxcoshsinxh+cosxsinhh)Regroup.=limh0(sinx(cosh1h)+cosx(sinhh))Factor outsinxandcosx. =sinx·0+cosx·1Apply trig limit formulas.=cosxSimplify.

Figure 3.27 shows the relationship between the graph of f(x)=sinxf(x)=sinx and its derivative f(x)=cosx.f(x)=cosx. Notice that at the points where f(x)=sinxf(x)=sinx has a horizontal tangent, its derivative f(x)=cosxf(x)=cosx takes on the value zero. We also see that where f(x)=sinxf(x)=sinx is increasing, f(x)=cosx>0f(x)=cosx>0 and where f(x)=sinxf(x)=sinx is decreasing, f(x)=cosx<0.f(x)=cosx<0.

The functions f(x) = sin x and f’(x) = cos x are graphed. It is apparent that when f(x) has a maximum or a minimum that f’(x) = 0.
Figure 3.27 Where f(x)f(x) has a maximum or a minimum, f(x)=0f(x)=0 that is, f(x)=0f(x)=0 where f(x)f(x) has a horizontal tangent. These points are noted with dots on the graphs.

Example 3.39

Differentiating a Function Containing sin x

Find the derivative of f(x)=5x3sinx.f(x)=5x3sinx.

Checkpoint 3.25

Find the derivative of f(x)=sinxcosx.f(x)=sinxcosx.

Example 3.40

Finding the Derivative of a Function Containing cos x

Find the derivative of g(x)=cosx4x2.g(x)=cosx4x2.

Checkpoint 3.26

Find the derivative of f(x)=xcosx.f(x)=xcosx.

Example 3.41

An Application to Velocity

A particle moves along a coordinate axis in such a way that its position at time tt is given by s(t)=2sintts(t)=2sintt for 0t2π.0t2π. At what times is the particle at rest?

Checkpoint 3.27

A particle moves along a coordinate axis. Its position at time tt is given by s(t)=3t+2costs(t)=3t+2cost for 0t2π.0t2π. At what times is the particle at rest?

Derivatives of Other Trigonometric Functions

Since the remaining four trigonometric functions may be expressed as quotients involving sine, cosine, or both, we can use the quotient rule to find formulas for their derivatives.

Example 3.42

The Derivative of the Tangent Function

Find the derivative of f(x)=tanx.f(x)=tanx.

Checkpoint 3.28

Find the derivative of f(x)=cotx.f(x)=cotx.

The derivatives of the remaining trigonometric functions may be obtained by using similar techniques. We provide these formulas in the following theorem.

Theorem 3.9

Derivatives of tanx,cotx,secx,tanx,cotx,secx, and cscxcscx

The derivatives of the remaining trigonometric functions are as follows:

ddx(tanx)=sec2xddx(tanx)=sec2x
(3.13)
ddx(cotx)=csc2xddx(cotx)=csc2x
(3.14)
ddx(secx)=secxtanxddx(secx)=secxtanx
(3.15)
ddx(cscx)=cscxcotx.ddx(cscx)=cscxcotx.
(3.16)

Example 3.43

Finding the Equation of a Tangent Line

Find the equation of a line tangent to the graph of f(x)=cotxf(x)=cotx at x=π4.x=π4.

Example 3.44

Finding the Derivative of Trigonometric Functions

Find the derivative of f(x)=cscx+xtanx.f(x)=cscx+xtanx.

Checkpoint 3.29

Find the derivative of f(x)=2tanx3cotx.f(x)=2tanx3cotx.

Checkpoint 3.30

Find the slope of the line tangent to the graph of f(x)=tanxf(x)=tanx at x=π6.x=π6.

Higher-Order Derivatives

The higher-order derivatives of sinxsinx and cosxcosx follow a repeating pattern. By following the pattern, we can find any higher-order derivative of sinxsinx and cosx.cosx.

Example 3.45

Finding Higher-Order Derivatives of y=sinxy=sinx

Find the first four derivatives of y=sinx.y=sinx.

Analysis

Once we recognize the pattern of derivatives, we can find any higher-order derivative by determining the step in the pattern to which it corresponds. For example, every fourth derivative of sin x equals sin x, so

d4dx4(sinx)=d8dx8(sinx)=d12dx12(sinx)==d4ndx4n(sinx)=sinxd5dx5(sinx)=d9dx9(sinx)=d13dx13(sinx)==d4n+1dx4n+1(sinx)=cosx.d4dx4(sinx)=d8dx8(sinx)=d12dx12(sinx)==d4ndx4n(sinx)=sinxd5dx5(sinx)=d9dx9(sinx)=d13dx13(sinx)==d4n+1dx4n+1(sinx)=cosx.

Checkpoint 3.31

For y=cosx,y=cosx, find d4ydx4.d4ydx4.

Example 3.46

Using the Pattern for Higher-Order Derivatives of y=sinxy=sinx

Find d74dx74(sinx).d74dx74(sinx).

Checkpoint 3.32

For y=sinx,y=sinx, find d59dx59(sinx).d59dx59(sinx).

Example 3.47

An Application to Acceleration

A particle moves along a coordinate axis in such a way that its position at time tt is given by s(t)=2sint.s(t)=2sint. Find v(π/4)v(π/4) and a(π/4).a(π/4). Compare these values and decide whether the particle is speeding up or slowing down.

Checkpoint 3.33

A block attached to a spring is moving vertically. Its position at time tt is given by s(t)=2sint.s(t)=2sint. Find v(5π6)v(5π6) and a(5π6).a(5π6). Compare these values and decide whether the block is speeding up or slowing down.

Section 3.5 Exercises

For the following exercises, find dydxdydx for the given functions.

175.

y = x 2 sec x + 1 y = x 2 sec x + 1

176.

y = 3 csc x + 5 x y = 3 csc x + 5 x

177.

y = x 2 cot x y = x 2 cot x

178.

y = x x 3 sin x y = x x 3 sin x

179.

y = sec x x y = sec x x

180.

y = sin x tan x y = sin x tan x

181.

y = ( x + cos x ) ( 1 sin x ) y = ( x + cos x ) ( 1 sin x )

182.

y = tan x 1 sec x y = tan x 1 sec x

183.

y = 1 cot x 1 + cot x y = 1 cot x 1 + cot x

184.

y = cos x ( 1 + csc x ) y = cos x ( 1 + csc x )

For the following exercises, find the equation of the tangent line to each of the given functions at the indicated values of x.x. Then use a calculator to graph both the function and the tangent line to ensure the equation for the tangent line is correct.

185.

[T] f(x)=sinx,x=0f(x)=sinx,x=0

186.

[T] f(x)=cscx,x=π2f(x)=cscx,x=π2

187.

[T] f(x)=1+cosx,x=3π2f(x)=1+cosx,x=3π2

188.

[T] f(x)=secx,x=π4f(x)=secx,x=π4

189.

[T] f(x)=x2tanx,x=0f(x)=x2tanx,x=0

190.

[T] f(x)=5cotx,x=π4f(x)=5cotx,x=π4

For the following exercises, find d2ydx2d2ydx2 for the given functions.

191.

y = x sin x cos x y = x sin x cos x

192.

y = sin x cos x y = sin x cos x

193.

y = x 1 2 sin x y = x 1 2 sin x

194.

y = 1 x + tan x y = 1 x + tan x

195.

y = 2 csc x y = 2 csc x

196.

y = sec 2 x y = sec 2 x

197.

Find all xx values on the graph of f(x)=−3sinxcosxf(x)=−3sinxcosx where the tangent line is horizontal.

198.

Find all xx values on the graph of f(x)=x2cosxf(x)=x2cosx for 0<x<2π0<x<2π where the tangent line has slope 2.

199.

Let f(x)=cotx.f(x)=cotx. Determine the points on the graph of ff for 0<x<2π0<x<2π where the tangent line(s) is (are) parallel to the line y=−2x.y=−2x.

200.

[T] A mass on a spring bounces up and down in simple harmonic motion, modeled by the function s(t)=−6costs(t)=−6cost where ss is measured in inches and tt is measured in seconds. Find the rate at which the spring is oscillating at t=5t=5 s.

201.

Let the position of a swinging pendulum in simple harmonic motion be given by s(t)=acost+bsints(t)=acost+bsint where aa and bb are constants, tt measures time in seconds, and ss measures position in centimeters. If the position is 0 cm and the velocity is 3 cm/s when t=0t=0, find the values of aa and bb.

202.

After a diver jumps off a diving board, the edge of the board oscillates with position given by s(t)=−5costs(t)=−5cost cm at tt seconds after the jump.

  1. Sketch one period of the position function for t0.t0.
  2. Find the velocity function.
  3. Sketch one period of the velocity function for t0.t0.
  4. Determine the times when the velocity is 0 over one period.
  5. Find the acceleration function.
  6. Sketch one period of the acceleration function for t0.t0.
203.

The number of hamburgers sold at a fast-food restaurant in Pasadena, California, is given by y=10+5sinxy=10+5sinx where yy is the number of hamburgers sold and xx represents the number of hours after the restaurant opened at 11 a.m. until 11 p.m., when the store closes. Find yy and determine the intervals where the number of burgers being sold is increasing.

204.

[T] The amount of rainfall per month in Phoenix, Arizona, can be approximated by y(t)=0.5+0.3cost,y(t)=0.5+0.3cost, where tt is months since January. Find yy and use a calculator to determine the intervals where the amount of rain falling is decreasing.

For the following exercises, use the quotient rule to derive the given equations.

205.

d d x ( cot x ) = csc 2 x d d x ( cot x ) = csc 2 x

206.

d d x ( sec x ) = sec x tan x d d x ( sec x ) = sec x tan x

207.

d d x ( csc x ) = csc x cot x d d x ( csc x ) = csc x cot x

208.

Use the definition of derivative and the identity

cos(x+h)=cosxcoshsinxsinhcos(x+h)=cosxcoshsinxsinh to prove that d(cosx)dx=sinx.d(cosx)dx=sinx.

For the following exercises, find the requested higher-order derivative for the given functions.

209.

d3ydx3d3ydx3 of y=3cosxy=3cosx

210.

d2ydx2d2ydx2 of y=3sinx+x2cosxy=3sinx+x2cosx

211.

d4ydx4d4ydx4 of y=5cosxy=5cosx

212.

d2ydx2d2ydx2 of y=secx+cotxy=secx+cotx

213.

d3ydx3d3ydx3 of y=x10secxy=x10secx

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