Question

differentiate the following function. y = x^7 e^x
\frac{dy}{dx}=square

Explanation:

Step1: Apply product - rule

The product - rule states that if $y = u\cdot v$, then $y^\prime=u^\prime v + uv^\prime$. Here, let $u = x^{7}$ and $v = e^{x}$.

Step2: Differentiate $u = x^{7}$

Using the power - rule $\frac{d}{dx}(x^{n})=nx^{n - 1}$, we have $u^\prime=\frac{d}{dx}(x^{7}) = 7x^{6}$.

Step3: Differentiate $v = e^{x}$

The derivative of $e^{x}$ with respect to $x$ is $e^{x}$, so $v^\prime=e^{x}$.

Step4: Substitute into product - rule

$y^\prime=(x^{7})^\prime e^{x}+x^{7}(e^{x})^\prime=7x^{6}e^{x}+x^{7}e^{x}=x^{6}e^{x}(7 + x)$.

Answer:

$x^{6}e^{x}(7 + x)$