Question

differentiate the following function.

$y = x^{10}e^{x}$

$\frac{dy}{dx} = square$

Explanation:

Step1: Apply product rule

The product rule states that if \(y = u\cdot v\), then \(y^\prime=u^\prime v + uv^\prime\). Let \(u = x^{10}\) and \(v = e^{x}\).
First, find \(u^\prime\): Using the power rule \((x^n)^\prime=nx^{n - 1}\), so \(u^\prime=\frac{d}{dx}(x^{10})=10x^{9}\).
Second, find \(v^\prime\): Using the rule \((e^{x})^\prime=e^{x}\), so \(v^\prime=\frac{d}{dx}(e^{x})=e^{x}\).

Step2: Substitute into product rule formula

\(\frac{dy}{dx}=u^\prime v+uv^\prime\)
Substitute \(u = x^{10}\), \(u^\prime = 10x^{9}\), \(v = e^{x}\), and \(v^\prime = e^{x}\) into the formula:
\(\frac{dy}{dx}=10x^{9}\cdot e^{x}+x^{10}\cdot e^{x}\)
Factor out \(x^{9}e^{x}\):
\(\frac{dy}{dx}=x^{9}e^{x}(10 + x)\)

Answer:

\(x^{9}e^{x}(x + 10)\)