5.1.3 The Derivatives of Trigonometric Functions

Question 1

derivatives of trigonometric functions

Answer 1

• If f ( x ) = sin x , f ′ ( x ) = cos x . If f ( x ) = cos x , f ′ ( x ) = − sin x .
• Use the derivatives of sine and cosine along with different differentiation techniques to find the derivatives of the other trigonometric functions.

Question 2

note

Answer 2

• It is not clear what the derivative of the sine function
  is when you apply the formula for the derivative.
• However, you can get a good idea what the graph of
  the derivative looks like by considering the way that
  the slopes of its tangent lines change.
• Notice that the tangent lines start with positive
  slopes. Then the slopes become negative. Then
  the slopes become positive again.
• If you plot the values of the slopes on a graph, you
  will trace out the cosine curve.
• The derivative of the sine function is the cosine
  function.
• The same process can be used on the cosine
  function. However, the results are a little
  unexpected.
• The derivative of the cosine function is the negative
  sine function.
• Find the derivatives of other trigonometric functions
  by expressing them in terms of sine and cosine and
  then applying different computational techniques.
• For example, the tangent function can be expressed
  as a quotient of the sine and cosine functions. So
  finding the derivative of the tangent function
  requires the quotient rule.
• The derivative of the tangent function is the square
  of the secant function.

Question 3

Find the derivative.

f (t) = 3t sec t

Answer 3

f ′(t) = 3 sec t (1 + t tan t)

Question 4

Find the derivative.

f (x) = cot^2 x

Answer 4

f ′(x) = −2 cot x csc^2 x

Question 5

Find the derivative.

f (t) = cos^2 t

Answer 5

f ′(t) = −2 cos t sin t

Question 6

Find the derivative.

f(x)=√2tanx/3

Answer 6

f′(x)=√2/3 sec^2x

Question 7

Find the derivative.

f (x) = sin x

Answer 7

f ′(x) = cos x

Question 8

Find the derivative.

f (t) = cos^4 t

Answer 8

f ′(t) = −4 cos^3 t sin t

Question 9

Find the derivative.

f (t) = tan^3 t

Answer 9

f ′(t) = 3 tan^2 t sec2 t

Question 10

Find the derivative.

f (x) = 3 sin x

Answer 10

f′(x)=3cosx

Question 11

Find the derivative.

f (t) = 3t sin^2 t

Answer 11

f ′(t) = 3 sin t (sin t + 2t cos t)

Question 12

Find the derivative.

f (x) = 4 tan x

Answer 12

f′(x)=4sec^2x

Question 13

Find the derivative.

f (x) = tan^2 x

Answer 13

f′(x)=2tanxsec^2x

Question 14

Find the derivative.

f(x)=−√6sinx

Answer 14

f′(x)=−√6cosx

Question 15

Find the derivative.

f (x) = sin x cos x

Answer 15

f′(x)=−sin^2x+cos^2x

Question 16

Find the derivative.

f(x)=cosx/2

Answer 16

f′(x)=−1/2sinx

Question 17

Find the derivative.

f (x) = sec x

Answer 17

f′(x)=tanxsecx