Question

webwork 2 - topics (1 point) find $\frac{dy}{dx}$ for the function $y = x^{5}cos(x)$. $\frac{dy}{dx}=square$

Explanation:

Step1: Apply product - rule

The product - rule states that if $y = u\cdot v$, then $\frac{dy}{dx}=u\frac{dv}{dx}+v\frac{du}{dx}$. Here, $u = x^{5}$ and $v=\cos(x)$.

Step2: Differentiate $u$ and $v$

The derivative of $u = x^{5}$ with respect to $x$ is $\frac{du}{dx}=5x^{4}$ (using the power - rule $\frac{d}{dx}(x^{n})=nx^{n - 1}$), and the derivative of $v=\cos(x)$ with respect to $x$ is $\frac{dv}{dx}=-\sin(x)$.

Step3: Substitute into product - rule

$\frac{dy}{dx}=x^{5}\cdot(-\sin(x))+\cos(x)\cdot5x^{4}=5x^{4}\cos(x)-x^{5}\sin(x)$

Answer:

$5x^{4}\cos(x)-x^{5}\sin(x)$