302
Chapter 3 | Derivatives
Example 3.62 Applying the Power Rule to a Rational Power
Find the equation of the line tangent to the graph of y = x 2/3 at x =8.
Solution First find
dy dx
and evaluate it at x =8. Since dy dx = 2 3
dy dx | x =8
x −1/3 and
= 1 3
the slope of the tangent line to the graph at x =8 is 1 3 . Substituting x =8 into the original function, we obtain y =4. Thus, the tangent line passes through the point (8, 4). Substituting into the point-slope formula for a line, we obtain the tangent line y = 1 3 x + 4 3 .
3.44
Find the derivative of s ( t ) = 2 t +1.
Derivatives of Inverse Trigonometric Functions We now turn our attention to finding derivatives of inverse trigonometric functions. These derivatives will prove invaluable in the study of integration later in this text. The derivatives of inverse trigonometric functions are quite surprising in that their derivatives are actually algebraic functions. Previously, derivatives of algebraic functions have proven to be algebraic functions and derivatives of trigonometric functions have been shown to be trigonometric functions. Here, for the first time, we see that the derivative of a function need not be of the same type as the original function. Example 3.63 Derivative of the Inverse Sine Function
Use the inverse function theorem to find the derivative of g ( x ) = sin −1 x .
Solution Since for x in the interval ⎡
⎤ ⎦ , f ( x ) = sin x is the inverse of g ( x ) = sin −1 x , begin by finding f ′( x ).
⎣ − π
π 2
2 ,
Since
⎛ ⎝ sin −1 x
⎞ ⎠ = 1− x 2 ,
f ′( x ) =cos x and f ′ ⎛
⎞ ⎠ =cos
⎝ g ( x )
we see that
g ′( x ) = d dx ⎛
⎞ ⎠ = 1 f ′ ⎛
⎝ sin −1 x
= 1
.
⎞ ⎠
⎝ g ( x )
1− x 2
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