Exam 3: Differentiation Rules

Differentiate the following functions: (a) $f ( t ) = \tan ^ { - 1 } \sqrt { 1 - t }$ (b) $f ( x ) = \sin ^ { - 1 } \sqrt { 2 x + t }$ (c) $g ( x ) = \sqrt { \tan ^ { - 1 } ( 3 x ) }$

If $x ^ { 3 } + y ^ { 3 } = 1$ , find an expression for $y ^ { \prime \prime }$ .

During the flood of 1997 in Fargo, North Dakota, the Red River of the North rose menacingly during the month of April. Suppose that the official river level at noon on day t is given by the function R(t). We know that R(April 9) = 35.5 feet and $R ^ { \prime }$ (April 9) = 1. (a) What are the units of $R ^ { \prime }$ (t)? (b) Construct a linear function to estimate the water level of the Red River for dates near April 9. (b) to estimate R(April 7) and R(April 11). (c) Use this model from part (d) The Red River crested on April 17 at 39.79 feet. What is $R ^ { \prime }$ (April 17)? (e) If the Red River level on April 22, R(April 22), was 39 feet and $R ^ { \prime }$ (April 22) was -0.3, use a linear model to estimate R(April 26).

Find $\frac { d y } { d x } .$ (a) $y = \left( x ^ { 2 } + 3 x + 2 \right) ^ { 4 }$ (b) $y = \sqrt [ 3 ] { 1 + x ^ { 3 } }$ (c) $y = x e ^ { 1 / x }$ (d) $y = \frac { 1 } { \left( x ^ { 2 } + 4 x + 1 \right) ^ { 3 } }$

Let $f ( x ) = x ^ { \frac { 1 } { x } }$ . Find the value of $f ^ { \prime } ( e )$ .

Suppose the amount of a drug left in the body t hours after administration is $\frac { 20 } { ( t + 1 ) }$ mg. In mg=h, find the rate of decrease of the drug 4 hours after administration.

Show that for sufficiently small values of h, $\sqrt { 4 + h } \approx 2 + \frac { h } { 4 }$

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

x f(x) (x) g(x) (x) -6 7 -8 -6 7 -4 1 -5 0 5 -2 3 -2 4 3 0 5 0 6 1 2 -5 1 6 -1 4 -3 3 4 -3 6 1 5 0 -5 -Find $\frac { d } { d x } \left( g ^ { 2 } ( x ) \right) \text { when } x = 2$

Find an equation of the line tangent to the curve $y = \frac { 4 } { 3 } x ^ { 3 } + 11 x ^ { 2 } + 10 x + 4 \text { at } x = - 3$

Find an equation of the tangent line to the curve $y ^ { 2 } - 3 x y + 2 x ^ { 2 } = 4$ at (3, 2).

What is the slope of the tangent line to the curve $x y ^ { 3 } + y - 1 = 0$ at the point (0, 1)?

If $f ( x ) = 3 x - x ^ { 2 }$ , find $f ^ { \prime } ( x )$ .

Find the y-intercept of the tangent to the ellipse $x ^ { 2 } + 3 y ^ { 2 } = 1$ at the point ( $\frac { 1 } { 2 } , \frac { 1 } { 2 }$ ).

Let $f ( x ) = x ^ { \sqrt { x } }$ . Find the value of $f ^ { \prime } ( 4 )$ .

Find $\frac { d y } { d x }$ .(a) $y = \ln \left( \cos ^ { 2 } x + 1 \right)$ (b) $y = \ln \ln x$

x f(x) (x) g(x) (x) -6 7 -8 -6 7 -4 1 -5 0 5 -2 3 -2 4 3 0 5 0 6 1 2 -5 1 6 -1 4 -3 3 4 -3 6 1 5 0 -5 -Find $\frac { d } { d x } \left( \frac { 2 \cdot g ( x ) } { x ^ { 2 } + 1 } \right) \text { when } x = 0$

Find the derivative of the following functions: (a) $f ( x ) = \frac { \sqrt [ 3 ] { x } + 2 } { \sqrt [ 3 ] { x } - 2 }$ (b) $f ( x ) = \frac { 1 } { x ^ { 2 } + 1 }$ (c) $f ( r ) = ( 1 + \sqrt { r } ) ( 1 - \sqrt { r } )$ (d) $f ( t ) = \frac { t e ^ { t } } { t ^ { 3 } + t }$

Let $f ( x ) = \frac { 4 } { 3 } x ^ { 3 } + 11 x ^ { 2 } + 10 x + 4$ . For what value of x is $f ^ { \prime } ( x ) = 3 ?$

Suppose that $F ( x ) = f ( g ( x ) )$ and $g ( 3 ) = 5$ , $g ^ { \prime } ( 3 ) = 3 \text {, }$ $f ^ { \prime } ( 3 ) = 1 , f ^ { \prime \prime } ( 5 ) = 4$ Find the value of $F ^ { \prime } ( 3 )$