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)