5.1.3 The Derivatives of Trigonometric Functions Flashcards Preview

AP Calculus AB > 5.1.3 The Derivatives of Trigonometric Functions > Flashcards

Flashcards in 5.1.3 The Derivatives of Trigonometric Functions Deck (17)
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1
Q

derivatives of trigonometric functions

A
  • 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.
2
Q

note

A
  • 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.
3
Q

Find the derivative.

f (t) = 3t sec t

A

f ′(t) = 3 sec t (1 + t tan t)

4
Q

Find the derivative.

f (x) = cot^2 x

A

f ′(x) = −2 cot x csc^2 x

5
Q

Find the derivative.

f (t) = cos^2 t

A

f ′(t) = −2 cos t sin t

6
Q

Find the derivative.

f(x)=√2tanx/3

A

f′(x)=√2/3 sec^2x

7
Q

Find the derivative.

f (x) = sin x

A

f ′(x) = cos x

8
Q

Find the derivative.

f (t) = cos^4 t

A

f ′(t) = −4 cos^3 t sin t

9
Q

Find the derivative.

f (t) = tan^3 t

A

f ′(t) = 3 tan^2 t sec2 t

10
Q

Find the derivative.

f (x) = 3 sin x

A

f′(x)=3cosx

11
Q

Find the derivative.

f (t) = 3t sin^2 t

A

f ′(t) = 3 sin t (sin t + 2t cos t)

12
Q

Find the derivative.

f (x) = 4 tan x

A

f′(x)=4sec^2x

13
Q

Find the derivative.

f (x) = tan^2 x

A

f′(x)=2tanxsec^2x

14
Q

Find the derivative.

f(x)=−√6sinx

A

f′(x)=−√6cosx

15
Q

Find the derivative.

f (x) = sin x cos x

A

f′(x)=−sin^2x+cos^2x

16
Q

Find the derivative.

f(x)=cosx/2

A

f′(x)=−1/2sinx

17
Q

Find the derivative.

f (x) = sec x

A

f′(x)=tanxsecx

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