### Home > APCALC > Chapter 12 > Lesson 12.2.2 > Problem12-81

12-81.

Write the derivative with respect to $x$ for each of the following functions. Simplify, factoring where possible.

1. $f ( x ) = \sqrt { x } e ^ { \sqrt { x } }$

$\frac{1}{2}x^{-1/2}e^{\sqrt{x}}+\sqrt{x}e^{\sqrt{x}}\Big(\frac{1}{2}x^{-1/2}\Big)$

$\frac{1}{2}x^{-1/2}e^{\sqrt{x}}+\frac{1}{2}e^{\sqrt{x}}$

1. $g(x) = \cos(kx) \sin(kx)$, $k$ is a constant

$-k\sin(kx)\sin(kx)+k\cos(kx)(\cos(kx)$

$k(\cos^2(x)-\sin^2(x))$

1. $h ( x ) = \frac { x ^ { 3 } } { \sqrt { \operatorname { ln } ( 1 + x ^ { 2 } ) } }$

$\frac{3x^2\sqrt{\ln(1+x^2)}-x^3\cdot\frac{1}{2}(\ln(1+x^2))^{-1/2}\cdot\frac{2x}{1+x^2}}{\ln(1+x^2)}$

1. $j ( x ) = \operatorname { tan } ( p _ { 0 } e ^ { - x ^ { 2 } } )$, $p$ is a constant

$\sec^2(p_0e^{-x^2})(-2xp_0e^{-x^2})$