Exam 3: Differentiation Rules

The period of a pendulum is given by the formula T = 2 $\pi$ $\sqrt { \frac { L } { g } }$ , where L is the length of the pendulum in feet, g = 32 ft/ $S ^ { 2 }$ is the acceleration due to gravity, and T is the length of one period in seconds. If the length of the pendulum is measured to be three feet long to within $\pm \frac { 1 } { 8 }$ inch, what is the approximate percentage error in the calculated period, T?

The function f is graphed below. Let $g ( x ) = f ( f ( x ) ) , h ( x ) = f \left( x ^ { 3 } \right)$ and $k ( x ) = [ f ( x ) ] ^ { 3 }$ .Use the graph to estimate each of the following.(a) $g ^ { \prime } ( 2 )$ (b) $h ^ { \prime } ( 2 )$ (c) $k ^ { \prime } ( 2 )$

Let $f ( x ) = \tan ^ { - 1 } \left( x ^ { 2 } + 1 \right)$ . Find the value of $f ^ { \prime } ( 1 )$ .

Find the minimum value of $f ( x ) = x ^ { - \frac { 1 } { x } }$ .

Find an equation of the tangent line to the curve $y = \frac { 1 - x } { 1 + x } \text { at } ( - 2 , - 3 )$

Differentiate the following functions: (a) $f ( x ) = x ^ { 3 } e ^ { x }$ (b) $g ( x ) = \frac { x ^ { 2 } + 1 } { x ^ { 2 } - 1 }$ (c) $h ( x ) = \frac { e ^ { x } } { e ^ { x } + 1 }$ (d) $\varnothing ( x ) = \left( 1 - \frac { 1 } { x } \right) ^ { 2 }$

Find an equation of the tangent line to the curve $2 x ^ { 3 } + 2 y ^ { 3 } - 9 x y = 0$ at the point (1, 2).

Find the value of the limit $\lim _ { x \rightarrow 0 } \frac { \sin 6 t } { \sin 4 t }$ .

Find the linear approximation to $f ( x ) = \frac { 1 } { \sqrt { x ^ { 3 } + 1 } }$ at $a = 2 .$

Let $f ( x ) = \log _ { 10 } \pi ^ { x }$ . Find the value of $f ^ { \prime } ( 100 )$ .

Let $f ( x ) = \sin ^ { - 1 } ( 2 x )$ . Find the value of $f ^ { \prime } ( 0 )$ .

Find the derivative of $f ( x ) = \frac { \tan x } { x ^ { 3 } }$ .

Use differentials to approximate $\sqrt { 26 }$ .

Find the derivative of $f ( x ) = \sin ^ { 3 } x$ .

If sin y = x, find the value of $\frac { d y } { d x }$ at the point $\left( \frac { 1 } { 2 } , \frac { \pi } { 6 } \right)$ .

The curve $y ^ { 2 } - x ^ { 2 } y - 6 = 0$ has two tangents at x = 1. What are their equations?

Find the slope of the tangent line to the curve $x ^ { 3 } + y ^ { 3 } = 6 x y$ at the point (3, 3).

Find an equation of the tangent to the curve $y = 2 \sin ^ { 2 } x \text { at } x = \frac { \pi } { 4 }$

Find the derivative of $f ( x ) = e ^ { x } \left( x ^ { 2 } + 2 \right)$ .

Find the linear approximation to $f ( x ) = \frac { 1 } { ( 2 + x ) ^ { 3 } }$ at $a = 0$