Derivative of exp(u) , exp(u(x))

Derivative of exp(u(x))

Let $u(x)$ be a function of the real variable $x$. The derivative $f’(x)$ of the function $f(x) = \exp(u(x))$ is given by:

$$ \forall x \in \mathbb{R}, \quad f’(x) = u’(x) \cdot \exp(u(x)) $$

Proof

Consider the function $g(x) = \exp(x)$ and the function $h(x) = u(x)$. Then, $f(x) = g(h(x))$. Using the chain rule of differentiation, we have:

$$ f’(x) = g’(h(x)) \cdot h’(x) $$

Using the derivative rule of the exponential function, we have:

$$ g’(x) = \exp(x) $$

Hence:

$$ g’(h(x)) = \exp(u(x)) $$

Using the derivative of the function $u(x)$, we have:

$$ h’(x) = u’(x) $$

Finally, we obtain:

$$ f’(x) = \exp(u(x)) \cdot u’(x) $$

Therefore, we have:

$$ \forall x \in \mathbb{R}, \quad f’(x) = u’(x) \cdot \exp(u(x)) $$