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)
First 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}


Where \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)

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