Chapter 3 | Derivatives
281
Example 3.42 The Derivative of the Tangent Function
Find the derivative of f ( x ) = tan x .
Solution Start by expressing tan x as the quotient of sin x and cos x : f ( x ) = tan x = sin x cos x . Now apply the quotient rule to obtain f ′( x ) = cos x cos x −(−sin x )sin x (cos x ) 2 . Simplifying, we obtain f ′( x ) = cos 2 x + sin 2 x cos 2 x . Recognizing that cos 2 x +sin 2 x =1, by the Pythagorean theorem, we now have f ′( x ) = 1 cos 2 x . Finally, use the identity sec x = 1 cos x to obtain f ′( x ) = sec 2 x .
Find the derivative of f ( x ) =cot x .
3.28
The derivatives of the remaining trigonometric functions may be obtained by using similar techniques. We provide these formulas in the following theorem.
Theorem 3.9: Derivatives of tan x , cot x , sec x , and csc x The derivatives of the remaining trigonometric functions are as follows:
d dx (tan d dx (cot d dx (sec d dx (csc
(3.13)
x ) = sec 2 x x ) =−csc 2 x
(3.14)
(3.15)
x ) = sec x tan x
(3.16)
x ) =−csc x cotx.
Example 3.43
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