Question

find the first and second derivative of the function.
$g(r) = \sqrt{r} + \sqrt9{r}$
$g(r) = \square$
$g(r) = \square$
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1. - / 4.54 points

find $\frac{d^{111}}{dx^{111}}(\sin x)$.
$\square$
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Explanation:

Step1: Rewrite function with exponents

$G(r) = r^{1/2} + r^{1/9}$

Step2: Compute first derivative

$G'(r) = \frac{1}{2}r^{-1/2} + \frac{1}{9}r^{-8/9}$

Step3: Compute second derivative

$G''(r) = -\frac{1}{4}r^{-3/2} - \frac{8}{81}r^{-17/9}$

Step4: Find 111th derivative of sinx

Cycle of sinx derivatives: 4. $111 \mod 4 = 3$, so $\frac{d^{111}}{dx^{111}}(\sin x) = -\cos x$

Answer:

$G'(r) = \frac{1}{2}r^{-1/2} + \frac{1}{9}r^{-8/9}$
$G''(r) = -\frac{1}{4}r^{-3/2} - \frac{8}{81}r^{-17/9}$
$\frac{d^{111}}{dx^{111}}(\sin x) = -\cos x$