Chapter 3 | Derivatives
283
Find the slope of the line tangent to the graph of f ( x ) = tan x at x = π 6 .
3.30
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 . Example 3.45 Finding Higher-Order Derivatives of y =sin x
Find the first four derivatives of y = sin x .
Solution Each step in the chain is straightforward:
y = sin x
dy dx = cos
x
d 2 y dx 2 d 3 y dx 3 d 4 y dx 4
= −sin x
= −cos x
= sin x .
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 d 4 dx 4 (sin x ) = d 8 dx 8 (sin x ) = d 12 dx 12 (sin x ) =…= d 4 n dx 4 n (sin x ) = sin x d 5 dx 5 (sin x ) = d 9 dx 9 (sin x ) = d 13 dx 13 (sin x ) =…= d 4 n +1 dx 4 n +1 (sin x ) =cos x .
d 4 y dx 4
3.31
For y =cos x , find
.
Example 3.46 Using the Pattern for Higher-Order Derivatives of y =sin x
74 dx 74
Find d
(sin x ).
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