Mathematics

Problem

(a) Let

g(x)

be a differentiable function. Show that the derivative of the composite function

E(x)=e^{g(x)}

E'(x)=g'(x)\,e^{g(x)}

. (b) Hence explain why a function of the form

g'(x)\,e^{g(x)}

always possesses an antiderivative of the form

e^{g(x)}+C

, where

C

is a constant of integration.

Intuitive explanation

The chain rule is your key tool here: when you differentiate a composite function, you multiply the derivative of the outer function by the derivative of the inner function. In part (a), the outer function is the exponential $e^{(\cdot)}$ , which has the special property that its derivative is itself. Part (b) then flips the logic—if differentiating $e^{g(x)}$ gives you $g'(x)e^{g(x)}$ , then integrating $g'(x)e^{g(x)}$ must undo that process and return you to $e^{g(x)}$ .

Intuitive explanation

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