Composite function

Introduction: Besides the sum, product and quotient of two functions, there also exists so-called composite functions. These are functions of which the output of the "interior function" is used as input for the "exterior function".

Definition: The function $y(x)$ is a composite function if $y(x)$ can be written as $u(v(x))$, where the output of $v(x)$ is used as the input for $u(v)$.

Example: The function $y(x) = e^{3x^2-1}$ can be written as $y(x) = u(v(x))$, with
$$v(x) = 3x^2-1 \quad \text{and} \quad u(v) = e^v.$$
Then the composite function prescription is given by
$$y(x) = u\big(v(x)\big) = e^{v(x)} = e^{3x^2-1}.$$

Remark: In many cases a function can be written as a composite function in more than one way. In the example above we could have chosen for $v(x) = x^2$ and $u(v) = e^{3v-1}$. Because of the chain rule it is useful to choose the functions $v(x)$ and $u(v)$ in such a way that the two functions are both differentiable by the use of the rules of differentiation.