Question

let $u(x)=sin(x)$ and $v(x)=x^{20}$ and $f(x)=\frac{u(x)}{v(x)}$. $u(x)=$ $v(x)=$ $f=\frac{uv - uv}{v^{2}}=$ question help: video message instructor

Explanation:

Step1: Find derivative of $u(x)$

The derivative of $\sin(x)$ with respect to $x$ is $\cos(x)$. So, $u'(x)=\cos(x)$.

Step2: Find derivative of $v(x)$

Using the power - rule $\frac{d}{dx}(x^n)=nx^{n - 1}$, for $v(x)=x^{20}$, $v'(x)=20x^{19}$.

Step3: Apply the quotient - rule

The quotient - rule for $f(x)=\frac{u(x)}{v(x)}$ is $f'(x)=\frac{u'(x)v(x)-u(x)v'(x)}{v(x)^2}$. Substituting $u(x)=\sin(x)$, $v(x)=x^{20}$, $u'(x)=\cos(x)$ and $v'(x)=20x^{19}$ into the quotient - rule formula, we get $f'(x)=\frac{\cos(x)\cdot x^{20}-\sin(x)\cdot20x^{19}}{(x^{20})^2}=\frac{x^{19}(x\cos(x)-20\sin(x))}{x^{40}}=\frac{x\cos(x)-20\sin(x)}{x^{21}}$.

Answer:

$u'(x)=\cos(x)$; $v'(x)=20x^{19}$; $f'(x)=\frac{x\cos(x)-20\sin(x)}{x^{21}}$