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

The derivative of exp(u) is given by: u’.exp(u). Let’s prove it using the derivative of exp and using the chain rule of differentiation.

## 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:

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