# Chain Rule for Derivatives

Composition of Two Functions

The composition $f \circ g$ of two functions $f$ and $g$ is defined as
$\left( {f \circ g} \right)\left( x \right) = f\left( {g\left( x \right)} \right)$

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:
$\begin{gathered}\left( {f \circ g} \right)\left( {x + \Delta x} \right) = f\left( {g\left( {x + \Delta x} \right)} \right) \\ \frac{d}{{dx}}\left[ {\left( {f \circ g} \right)\left( x \right)} \right] = \mathop {\lim }\limits_{\Delta x \to 0} \frac{{\left( {f \circ g} \right)\left( {x + \Delta x} \right) – \left( {f \circ g} \right)\left( x \right)}}{{\Delta x}} \\ \Rightarrow \frac{d}{{dx}}\left[ {\left( {f \circ g} \right)\left( x \right)} \right] = \mathop {\lim }\limits_{\Delta x \to 0} \frac{{f\left( {g\left( {x + \Delta x} \right) – g\left( x \right) + g\left( x \right)} \right) – f\left( {g\left( x \right)} \right)}}{{\Delta x}} \\ \Rightarrow \frac{d}{{dx}}\left[ {\left( {f \circ g} \right)\left( x \right)} \right] = \mathop {\lim }\limits_{\Delta x \to 0} \frac{{f\left( {g\left( x \right) + g\left( {x + \Delta x} \right) – g\left( x \right)} \right) – f\left( {g\left( x \right)} \right)}}{{\Delta x}} \\ \Rightarrow \frac{d}{{dx}}\left[ {\left( {f \circ g} \right)\left( x \right)} \right] = \mathop {\lim }\limits_{\Delta x \to 0} \frac{{f\left( {g\left( x \right) + g\left( {x + \Delta x} \right) – g\left( x \right)} \right) – f\left( {g\left( x \right)} \right)}}{{g\left( {x + \Delta x} \right) – g\left( x \right)}} \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \times \frac{{g\left( {x + \Delta x} \right) – g\left( x \right)}}{{\Delta x}} \\ \Rightarrow \frac{d}{{dx}}\left[ {\left( {f \circ g} \right)\left( x \right)} \right] = \mathop {\lim }\limits_{\Delta x \to 0} \frac{{f\left( {g\left( x \right) + g\left( {x + \Delta x} \right) – g\left( x \right)} \right) – f\left( {g\left( x \right)} \right)}}{{g\left( {x + \Delta x} \right) – g\left( x \right)}} \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \times \mathop {\lim }\limits_{\Delta x \to 0} \frac{{g\left( {x + \Delta x} \right) – g\left( x \right)}}{{\Delta x}} \\ \Rightarrow \frac{d}{{dx}}\left[ {\left( {f \circ g} \right)\left( x \right)} \right] = \mathop {\lim }\limits_{\Delta k \to 0} \frac{{f\left( {g\left( x \right) + \Delta k} \right) – f\left( {g\left( x \right)} \right)}}{{\Delta k}} \times \mathop {\lim }\limits_{\Delta x \to 0} \frac{{g\left( {x + \Delta x} \right) – g\left( x \right)}}{{\Delta x}} \\ \Rightarrow \frac{d}{{dx}}\left[ {\left( {f \circ g} \right)\left( x \right)} \right] = f’\left[ {g\left( x \right)} \right]g’\left( x \right) \\ \end{gathered}$

Here $\Delta k = g\left( {x + \Delta x} \right) – g\left( x \right) \to 0$ as $\Delta x \to 0$
${\left( {f \circ g} \right)^\prime }\left( x \right) = f’\left[ {g\left( x \right)} \right] \cdot g’\left( x \right)$