Chapter 3 | Derivatives
305
f ′( x ) = 2 x
.
1+ x 4
Example 3.66 Applying Differentiation Formulas to an Inverse Sine Function
Find the derivative of h ( x ) = x 2 sin −1 x .
Solution By applying the product rule, we have
h ′( x ) =2 x sin −1 x + 1
· x 2 .
1− x 2
Find the derivative of h ( x ) =cos −1 (3 x −1).
3.46
Example 3.67 Applying the Inverse Tangent Function
The position of a particle at time t is given by s ( t ) = tan −1 ⎛ ⎝ 1 t ⎞
⎠ for t ≥ 1
2 . Find the velocity of the particle at
time t =1.
Solution Begin by differentiating s ( t ) in order to find v ( t ). Thus, v ( t ) = s ′( t ) = 1 1+ ⎛ ⎝ 1 t
2 · −1 t 2
.
⎞ ⎠
Simplifying, we have
v ( t ) = − 1
.
t 2 +1
Thus, v (1) = − 1 2 .
Find the equation of the line tangent to the graph of f ( x ) = sin −1 x at x =0.
3.47
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