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Find the derivative of f ( x ) = 2 tan x 3 cot x .

f ( x ) = 2 sec 2 x + 3 csc 2 x

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Find the slope of the line tangent to the graph of f ( x ) = tan x at x = π 6 .

4 3

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Higher-order derivatives

The higher-order derivatives of sin x and cos x follow a repeating pattern. By following the pattern, we can find any higher-order derivative of sin x and cos x .

Finding higher-order derivatives of y = sin x

Find the first four derivatives of y = sin x .

Each step in the chain is straightforward:

y = sin x d y d x = cos x d 2 y d x 2 = sin x d 3 y d x 3 = cos x d 4 y d x 4 = sin x .
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For y = cos x , find d 4 y d x 4 .

cos x

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Using the pattern for higher-order derivatives of y = sin x

Find d 74 d x 74 ( sin x ) .

We can see right away that for the 74th derivative of sin x , 74 = 4 ( 18 ) + 2 , so

d 74 d x 74 ( sin x ) = d 72 + 2 d x 72 + 2 ( sin x ) = d 2 d x 2 ( sin x ) = sin x .
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For y = sin x , find d 59 d x 59 ( sin x ) .

cos x

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An application to acceleration

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

First find v ( t ) = s ( t ) :

v ( t ) = s ( t ) = cos t .

Thus,

v ( π 4 ) = 1 2 .

Next, find a ( t ) = v ( t ) . Thus, a ( t ) = v ( t ) = sin t and we have

a ( π 4 ) = 1 2 .

Since v ( π 4 ) = 1 2 < 0 and a ( π 4 ) = 1 2 > 0 , we see that velocity and acceleration are acting in opposite directions; that is, the object is being accelerated in the direction opposite to the direction in which it is travelling. Consequently, the particle is slowing down.

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A block attached to a spring is moving vertically. Its position at time t is given by s ( t ) = 2 sin t . Find v ( 5 π 6 ) and a ( 5 π 6 ) . Compare these values and decide whether the block is speeding up or slowing down.

v ( 5 π 6 ) = 3 < 0 and a ( 5 π 6 ) = −1 < 0 . The block is speeding up.

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Key concepts

  • We can find the derivatives of sin x and cos x by using the definition of derivative and the limit formulas found earlier. The results are
    d d x sin x = cos x d d x cos x = sin x .
  • With these two formulas, we can determine the derivatives of all six basic trigonometric functions.

Key equations

  • Derivative of sine function
    d d x ( sin x ) = cos x
  • Derivative of cosine function
    d d x ( cos x ) = sin x
  • Derivative of tangent function
    d d x ( tan x ) = sec 2 x
  • Derivative of cotangent function
    d d x ( cot x ) = csc 2 x
  • Derivative of secant function
    d d x ( sec x ) = sec x tan x
  • Derivative of cosecant function
    d d x ( csc x ) = csc x cot x

For the following exercises, find d y d x for the given functions.

y = x 2 sec x + 1

d y d x = 2 x sec x tan x

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y = x 2 cot x

d y d x = 2 x cot x x 2 csc 2 x

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y = sec x x

d y d x = x sec x tan x sec x x 2

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

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

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y = 1 cot x 1 + cot x

d y d x = 2 csc 2 x ( 1 + cot x ) 2

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

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

y = x
The graph shows negative sin(x) and the straight line T(x) with slope −1 and y intercept 0.

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[T] f ( x ) = csc x , x = π 2

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[T] f ( x ) = 1 + cos x , x = 3 π 2

y = x + 2 3 π 2
The graph shows the cosine function shifted up one and has the straight line T(x) with slope 1 and y intercept (2 – 3π)/2.

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[T] f ( x ) = sec x , x = π 4

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[T] f ( x ) = x 2 tan x x = 0

y = x
The graph shows the function as starting at (−1, 3), decreasing to the origin, continuing to slowly decrease to about (1, −0.5), at which point it decreases very quickly.

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[T] f ( x ) = 5 cot x x = π 4

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For the following exercises, find d 2 y d x 2 for the given functions.

y = x sin x cos x

3 cos x x sin x

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y = x 1 2 sin x

1 2 sin x

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y = 2 csc x

csc ( x ) ( 3 csc 2 ( x ) 1 + cot 2 ( x ) )

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

( 2 n + 1 ) π 4 , where n is an integer

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

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

( π 4 , 1 ) , ( 3 π 4 , −1 )

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

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Let the position of a swinging pendulum in simple harmonic motion be given by s ( t ) = a cos t + b sin t . Find the constants a and b such that when the velocity is 3 cm/s, s = 0 and t = 0 .

a = 0 , b = 3

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

  1. Sketch one period of the position function for t 0 .
  2. Find the velocity function.
  3. Sketch one period of the velocity function for t 0 .
  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 t 0 .
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The number of hamburgers sold at a fast-food restaurant in Pasadena, California, is given by y = 10 + 5 sin x where y is the number of hamburgers sold and x represents the number of hours after the restaurant opened at 11 a.m. until 11 p.m., when the store closes. Find y and determine the intervals where the number of burgers being sold is increasing.

y = 5 cos ( x ) , increasing on ( 0 , π 2 ) , ( 3 π 2 , 5 π 2 ) , and ( 7 π 2 , 12 )

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

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

d d x ( cot x ) = csc 2 x

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

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

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Use the definition of derivative and the identity

cos ( x + h ) = cos x cos h sin x sin h to prove that d ( cos x ) d x = sin x .

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

d 3 y d x 3 of y = 3 cos x

3 sin x

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d 2 y d x 2 of y = 3 sin x + x 2 cos x

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d 4 y d x 4 of y = 5 cos x

5 cos x

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d 2 y d x 2 of y = sec x + cot x

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d 3 y d x 3 of y = x 10 sec x

720 x 7 5 tan ( x ) sec 3 ( x ) tan 3 ( x ) sec ( x )

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Source:  OpenStax, Calculus volume 1. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11964/1.2
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