differentiate the following function.\n\n$y = x^{10}e^{x}$\n\n$\frac{dy}{dx} = square$

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))