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):
1

derivatives of trigonometric functions

- 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

note

- 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

Find the derivative.
f (t) = 3t sec t

f ′(t) = 3 sec t (1 + t tan t)

4

Find the derivative.
f (x) = cot^2 x

f ′(x) = −2 cot x csc^2 x

5

Find the derivative.
f (t) = cos^2 t

f ′(t) = −2 cos t sin t

6

Find the derivative.
f(x)=√2tanx/3

f′(x)=√2/3 sec^2x

7

Find the derivative.
f (x) = sin x

f ′(x) = cos x

8

Find the derivative.
f (t) = cos^4 t

f ′(t) = −4 cos^3 t sin t

9

Find the derivative.
f (t) = tan^3 t

f ′(t) = 3 tan^2 t sec2 t

10

Find the derivative.
f (x) = 3 sin x

f′(x)=3cosx

11

Find the derivative.
f (t) = 3t sin^2 t

f ′(t) = 3 sin t (sin t + 2t cos t)

12

Find the derivative.
f (x) = 4 tan x

f′(x)=4sec^2x

13

Find the derivative.
f (x) = tan^2 x

f′(x)=2tanxsec^2x

14

Find the derivative.
f(x)=−√6sinx

f′(x)=−√6cosx

15

Find the derivative.
f (x) = sin x cos x

f′(x)=−sin^2x+cos^2x

16

Find the derivative.
f(x)=cosx/2

f′(x)=−1/2sinx

17

Find the derivative.
f (x) = sec x

f′(x)=tanxsecx

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