# Chain Rule for Derivatives

Composition of Two Functions

The composition $f \circ g$ of two functions $f$ and $g$ is defined as

Chain Rule

If $g$ is differentiable at the point $x$ and $f$ is differentiable at the point $g\left( x \right)$, then the composition $f \circ g$ of these functions is differentiable at $x$ and ${\left( {f \circ g} \right)^\prime }\left( x \right) = f'\left[ {g\left( x \right)} \right] \cdot g'\left( x \right)$.

Since $\left( {f \circ g} \right)\left( x \right) = f\left( {g\left( x \right)} \right)$, we take the increment or small change in the function:

Here $\Delta k = g\left( {x + \Delta x} \right) - g\left( x \right) \to 0$ as $\Delta x \to 0$