Question

Find the derivatives of the functions: $f\left(x\right)=3x^2{e}^{-4x}$

Short answer

The required answer is$-12x^2{e}^{-4x}+6x{e}^{-4x}$.

Step-by-step solution

Step 1. Given information.

The given function is:$f\left(x\right)=3x^2{e}^{-4x}$

Step 2. Find the derivative of the given function.

$$\begin{gathered} f\left(x\right)=3x^2{e}^{-4x} \\ =\frac{d}{dx}\left(3x^2{e}^{-4x}\right) \\ =3x^2\frac{d}{dx}\left(e^{-4x}\right)+e^{-4x}\frac{d}{dx}\left(3x^2\right) \\ =3x^2{e}^{-4x}\left(-4\right)+e^{-4x}6x \\ =-12x^2{e}^{-4x}+6x{e}^{-4x} \end{gathered}$$