CPM Homework Help : CALC Problem 12-92

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12-92.

Find the derivative with respect to $x$ of each of the following functions and simplify, factoring where possible.

$f ( x ) = \sqrt { x } e ^ { \sqrt { x } }$
Step 1(a):
$\frac{1}{2}x^{-1/2}e^{\sqrt{x}}+\sqrt{x}e^{\sqrt{x}}\Big(\frac{1}{2}x^{-1/2}\Big)$
Step 2(a):
$\frac{1}{2}x^{-1/2}e^{\sqrt{x}}+\frac{1}{2}e^{\sqrt{x}}$

$f(x) =\operatorname{cos} kx · \operatorname{sin} kx$
Step 1(b):
$-k\sin(kx)\sin(kx)+k\cos(kx)(\cos(kx)$
Step 2(b):
$k(\cos^2(x)-\sin^2(x))$

$f ( x ) = \frac { x ^ { 3 } } { \sqrt { \operatorname { ln } ( 1 + x ^ { 2 } ) } }$
Step (c):
$\frac{3x^2\sqrt{\ln(1+x^2)}-x^3\cdot\frac{1}{2}(\ln(1+x^2))^{-1/2}\cdot\frac{2x}{1+x^2}}{\ln(1+x^2)}$

$f ( x ) = \operatorname { tan } ( p _ { 0 } e ^ { - x ^ { 2 } } )$
Step (d):
$\sec^2(p_0e^{-x^2})(-2xp_0e^{-x^2})$

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