Question

1. find the derivatives of the following functions:

a. (5 pts.) $f(x)=sin(x)cos(x)$
b. (5 pts.) $g(x)=\frac{x}{sin(x)}$

Explanation:

Step1: Recall product - rule for A

The product - rule states that if $y = u(x)v(x)$, then $y^\prime=u^\prime(x)v(x)+u(x)v^\prime(x)$. For $f(x)=\sin(x)\cos(x)$, let $u = \sin(x)$ and $v=\cos(x)$. Then $u^\prime=\cos(x)$ and $v^\prime=-\sin(x)$.
$f^\prime(x)=\cos(x)\cos(x)+\sin(x)(-\sin(x))=\cos^{2}(x)-\sin^{2}(x)$.

Step2: Recall quotient - rule for B

The quotient - rule states that if $y=\frac{u(x)}{v(x)}$ ($v(x)
eq0$), then $y^\prime=\frac{u^\prime(x)v(x)-u(x)v^\prime(x)}{v^{2}(x)}$. For $g(x)=\frac{x}{\sin(x)}$, let $u = x$ and $v=\sin(x)$. Then $u^\prime = 1$ and $v^\prime=\cos(x)$.
$g^\prime(x)=\frac{1\times\sin(x)-x\times\cos(x)}{\sin^{2}(x)}=\frac{\sin(x)-x\cos(x)}{\sin^{2}(x)}$.

Answer:

A. $f^\prime(x)=\cos^{2}(x)-\sin^{2}(x)$
B. $g^\prime(x)=\frac{\sin(x)-x\cos(x)}{\sin^{2}(x)}$