How do I prove the Chain rule for derivatives?

Hint: Here, we have to prove the chain rule of derivatives. We will state the chain rule and then by using the limit definition for a function and the limit definition for the composite function we will prove the chain rule for derivatives. Chain rule for derivatives is used to find the derivative of a composite function.

Formula Used:
Limit definition is given by \[{f^{'}}\left( a \right) = \mathop {\lim }\limits_{h \to 0} \dfrac{{f\left( {a + h} \right) - f\left( a \right)}}{h}\].

Complete Step by Step Solution:
Chain rule is stated as if \[f\left( a \right)\] and \[g\left( a \right)\] are both differentiable functions and we define \[F\left( a \right) = \left( {f \circ g} \right)\left( a \right)\], then the derivative of \[F\left( a \right)\] is \[F'\left( a \right) = f'\left( {g\left( a \right)} \right)g'\left( a \right)\].
Now, we will prove the chain rule of derivatives by using the limit definition.
Limit definition is given by \[{f^{'}}\left( a \right) = \mathop {\lim }\limits_{h \to 0} \dfrac{{f\left( {a + h} \right) - f\left( a \right)}}{h}\] .
Now, we will find the derivative of the \[\left( {f \circ g} \right)'\left( x \right)\] by using the limit definition, we get
\[ \Rightarrow \left( {f \circ g} \right)'\left( a \right) = \mathop {\lim }\limits_{h \to 0} \dfrac{{f\left( {g\left( {a + h} \right)} \right) - f\left( {g\left( a \right)} \right)}}{h}\] .
Now, we will rewrite the equation, we get
\[ \Rightarrow \left( {f \circ g} \right)'\left( a \right) = \mathop {\lim }\limits_{h \to 0} \dfrac{{f\left( {g\left( a \right) + \Delta h} \right) - f\left( {g\left( a \right)} \right)}}{h}\] …………………………………………………….\[\left( 1 \right)\]
where \[\Delta h = g\left( {a + h} \right) - g\left( a \right)\]
Now, the equation is similar to the definition of \[f'\left( {g\left( a \right)} \right)\], so we get
\[ \Rightarrow f'\left( {g\left( a \right)} \right) = \mathop {\lim }\limits_{\Delta \to 0} \dfrac{{f\left( {g\left( a \right) + \Delta } \right) - f\left( {g\left( a \right)} \right)}}{\Delta }\]
Here \[\Delta \] means a small quantity tending to zero at different rates.
Now, we will rewrite the equation according to the limit definition, we get
\[ \Rightarrow \left( {f \circ g} \right)'\left( a \right) = \mathop {\lim }\limits_{h \to 0} \left( {\dfrac{{f\left( {g\left( a \right) + \Delta h} \right) - f\left( {g\left( a \right)} \right)}}{{\Delta h}} \cdot \dfrac{{\Delta h}}{h}} \right)\]
Now, by distributing inside the parenthesis, we get
\[ \Rightarrow \left( {f \circ g} \right)'\left( a \right) = \mathop {\lim }\limits_{h \to 0} \dfrac{{f\left( {g\left( a \right) + \Delta h} \right) - f\left( {g\left( a \right)} \right)}}{{\Delta h}} \cdot \mathop {\lim }\limits_{h \to 0} \dfrac{{\Delta h}}{h}\]
Now, by changing the limit using the limit definition, we get
\[ \Rightarrow \left( {f \circ g} \right)'\left( a \right) = \mathop {\lim }\limits_{\Delta h \to 0} \dfrac{{f\left( {g\left( a \right) + \Delta h} \right) - f\left( {g\left( a \right)} \right)}}{{\Delta h}} \cdot \mathop {\lim }\limits_{h \to 0} \dfrac{{\Delta h}}{h}\]
Now, by using \[\Delta h = f\left( {g\left( {a + h} \right)} \right) - g\left( a \right)\], we get
\[ \Rightarrow \left( {f \circ g} \right)'\left( a \right) = \mathop {\lim }\limits_{\Delta h \to 0} \dfrac{{f\left( {g\left( a \right) + \Delta h} \right) - f\left( {g\left( a \right)} \right)}}{{\Delta h}} \cdot \mathop {\lim }\limits_{h \to 0} \dfrac{{f\left( {g\left( {a + h} \right) - g\left( a \right)} \right)}}{h}\]
Now, by using the limit definition, we get
\[ \Rightarrow \left( {f \circ g} \right)'\left( a \right) = f'\left( {g\left( a \right)} \right) \cdot g'\left( a \right)\]
Chain rule is stated as if \[f\left( a \right)\] and\[g\left( a \right)\] are both differentiable functions and we define \[F\left( a \right) = \left( {f \circ g} \right)\left( a \right)\], then the derivative of \[F\left( a \right)\] is \[F'\left( a \right) = f'\left( {g\left( a \right)} \right)g'\left( a \right)\].
Therefore, the chain rule for derivatives is proved.

Note:
We know that Differentiation is a method of finding the derivative of the function and finding the rate of change of a function with respect to one variable. The reverse process is called anti differentiation. The limit definition is used to find the derivative of the function. The chain rule is also called the function of a function rule. While using the chain rule, we will differentiate the outer function and then we will differentiate the inner function.