Exam 4: More About Derivatives

Let $f ( x ) = \tan \left( 5 ^ { x } \right)$ Then $f ^ { \prime } ( x )$ is

Let $f ( x ) = \sin ( \cos x )$ Then $f ^ { \prime } ( x )$ is

The linear approximation $L ( x )$ to $f ( x ) = \sin x$ near $x _ { 0 } = - \frac { \pi } { 6 }$ is

Let $f ( x ) = \tan ^ { - 1 } ( \cos x )$ Then $f ^ { \prime } ( x )$ is

Let $y = x ^ { \ell ^ { x ^ { 2 } } }$ Then $y ^ { \prime }$ is

Let $f ( x ) = \mathrm { e } ^ { \cos ( 3 x ) }$ Then $f ^ { \prime } ( x )$ is

Let $y = \sin ( \ln x )$ Then $y ^ { \prime }$ is

Let $f ( x ) = \sinh \left( e ^ { x } \right)$ Then $f ^ { \prime } ( x )$ is

Let $f ( x ) = \tanh ^ { - 1 } \left( e ^ { x } \right)$ Then $f ^ { \prime } ( x )$ is

Using implicit differentiation on $y = x \sin y , y ^ { \prime }$ is

Let $y = \ln \left( \tan ^ { - 1 } x \right)$ Then $y ^ { \prime }$ is

Let $y = 2 ^ { \tan x }$ Then dy is

The linear approximation $L ( x )$ to $f ( x ) = \sqrt { x }$ near $x _ { 0 } = \frac { 1 } { 4 }$ is

Let $f ( x ) = \tan ^ { - 1 } \left( 3 ^ { x } \right)$ Then $f ^ { \prime } ( x )$ is

Let $y = \sin \left( 7 ^ { x } \right)$ Then dy is

Letting $f ( x ) = \tan x$ and $x _ { 0 } = \frac { \pi } { 4 }$ the approximation of tan 44° by differentials is

Let $y = x ^ {e ^ { x}}$ Then $y ^ { \prime }$ is

Using implicit differentiation on $2 x ^ { 2 } - x y + y ^ { 2 } = 4 , y$ is

Let $f ( x ) = \cosh ^ { - 1 } \left( 2 ^ { x } \right)$ Then $f ^ { \prime } ( x )$ is

Let $y = 5 \left( x ^ { 2 } - x + 1 \right) ^ { 100 }$ Then $y ^ { \prime }$ is