Exam 3: Short-Cuts to Differentiation

At what point(s)in the interval $[ 0,2 \pi ]$ , does the curve given by $y = 7 \ln ( \sin x )$ have horizontal tangents? Round to two decimal places.

Find the equation of the tangent line to $f ( x ) = e ^ { - 2 x }$ at x = 3 and use it to approximate the value of f(3.2).Round to 5 decimal places.

Differentiate $y = \sinh ( \ln ( 4 x ) )$ .

A table of values for a function F near x = 3 and tables of values for a function G near x = 3 and near x = 7 are given below.If $H ( x ) = G ( F ( x ) )$ , estimate H'(3)using the chain rule.Use right-hand estimates for $F ^ { \prime }$ and $G ^ { \prime }$ . x 2.9 3.0 3.1 F(x) 6.7 7.0 7.3 G(x) 5.2 5.0 4.8 x 6.9 7.0 7.1 G(x) 5.85 6.00 6.15

Differentiate $y = x \arctan ( 3 x )$ .

If $P ( w ) = 7 ^ { w } w ^ { - \frac { 7 } { 2 } }$ , then $P ^ { \prime } ( w ) = 7 ^ { w } \cdot \ln 7 \cdot w ^ { - \frac { 7 } { 2 } } + 7 ^ { w } \left( \frac { 7 } { 2 } w ^ { - \frac { 9 } { 2 } } \right)$ .

Find the derivative of $f ( w ) = \sin \left( 4 w ^ { 2 } \right) + \cos \left( 4 w ^ { 2 } \right)$ .

The derivative of $\cos \left( x ^ { 2 } + 8 \right)$ is $- 2 \sin \left( x ^ { 2 } + 8 \right)$ .

Find the slope of the line tangent to $y = e ^ { x } \left( 5 - x ^ { 2 } \right)$ when x = 2.Round to two decimal places.

A particle moves in such a way that $x ( t ) = 2 t ^ { 2 } + 3 \sin t$ , where x is the horizontal distance the particle has traveled, in units, and t is time, in seconds.What is the average rate of change between t = 0 and $t = \frac { \pi } { 2 }$ ? Specify units

The graph of f(x)is shown below.Suppose we want to estimate f (1.5)by using tangent line approximations at x = 1 and 2.Which tangent line yields the best approximation?

The derivative of $e ^ { e ^ { 4 x } }$ is $4 e ^ { 4 x + e ^ { 4 x } }$ .

Find $y ^ { \prime }$ when $4 x \sin y + 5 y \ln x = 11$

If $f ^ { \prime } ( x ) = \ln \left( x + 8 + e ^ { - 5 x } \right)$ , find $f ^ { \prime } ( 0 )$ .

Find the derivative of $f ( x ) = 5 ^ { 4 x + 4 }$ .

Find the derivative of $r ( \theta ) = 5 \cos ( \cos ( \cos ( \theta ) ) )$

If P dollars are invested at an annual rate of r%, then in t years this investment grows to F dollars, where $F = P \left( 1 + \frac { r } { 100 } \right) ^ { t }$ .Assuming P and r are constant, find $\frac { d F } { d t }$ .

The speed of a car at time t, 5 $\le$ t $\le$ 20, is given by $f ( t ) = - 2 ( t - 10 ) ^ { 2 }$ .The Mean Value Theorem guarantees that there will be a point in the interval [5, 20] when the instantaneous acceleration is equal to what number?

Find the derivative of $y=\frac{8 t^{3}-t^{2}+5}{t^{2}}$ .

A man plans to propose to a woman in romantic fashion by taking her up in an air balloon.Unfortunately, he pulls the diamond ring from his pocket and drops it over the side of the balloon's basket.The ring's position above the earth t seconds after it falls is given by the function $s ( t ) = - 16 t ^ { 2 } + 1325$ feet.How fast is the ring falling at the instant it hits the ground? 1325