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