Derivative of $e^{- 4 x}$

Find $\frac{d}{dx} \left(e^{- 4 x}\right)$.

The function $e^{- 4 x}$ is the composition $f{\left(g{\left(x \right)} \right)}$ of two functions $f{\left(u \right)} = e^{u}$ and $g{\left(x \right)} = - 4 x$.

Apply the chain rule $\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$:

The derivative of the exponential is $\frac{d}{du} \left(e^{u}\right) = e^{u}$:

Return to the old variable:

Apply the constant multiple rule $\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$ with $c = -4$ and $f{\left(x \right)} = x$:

Apply the power rule $\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$ with $n = 1$, in other words, $\frac{d}{dx} \left(x\right) = 1$:

Thus, $\frac{d}{dx} \left(e^{- 4 x}\right) = - 4 e^{- 4 x}$.

$\frac{d}{dx} \left(e^{- 4 x}\right) = - 4 e^{- 4 x}$A