Question

find the derivative of the function.
y = xe^{-5x}-4e^{x^{2}}
\frac{dy}{dx}=square
(type an exact answer in terms of e.)

Explanation:

Step1: Apply sum - difference rule

The derivative of a sum/difference of functions is the sum/difference of their derivatives. So, $\frac{dy}{dx}=\frac{d}{dx}(xe^{-5x})-\frac{d}{dx}(4e^{x^{2}})$.

Step2: Use product rule for $\frac{d}{dx}(xe^{-5x})$

The product rule states that if $u(x)$ and $v(x)$ are functions, then $(uv)' = u'v+uv'$. Let $u = x$ and $v=e^{-5x}$. Then $u'=1$ and $v'=- 5e^{-5x}$. So, $\frac{d}{dx}(xe^{-5x})=1\times e^{-5x}+x\times(-5e^{-5x})=e^{-5x}-5xe^{-5x}$.

Step3: Use chain - rule for $\frac{d}{dx}(4e^{x^{2}})$

The chain - rule states that if $y = f(g(x))$, then $y'=f'(g(x))\cdot g'(x)$. For $y = 4e^{x^{2}}$, let $u = x^{2}$, then $y = 4e^{u}$. $\frac{dy}{du}=4e^{u}$ and $\frac{du}{dx}=2x$. So, $\frac{d}{dx}(4e^{x^{2}})=4e^{x^{2}}\times2x = 8xe^{x^{2}}$.

Step4: Combine results

$\frac{dy}{dx}=e^{-5x}-5xe^{-5x}-8xe^{x^{2}}$.

Answer:

$e^{-5x}-5xe^{-5x}-8xe^{x^{2}}$