Derivative of a function of functions (Chain Rule)
In calculus we often end up having to differentiate a function of functions
The solution is arrived at using what is known as the Chain Rule, which states to take the derivative of the outside function and multiply it by the derivative of the inside function:
The proof is very easy, we start from basic principles and look first at the derivative of g(x):
Now, for the outer function:
Let’s make a variable substitution:
Following the same line of thinking as we did for the derivative of g:
Substituting the values back in, we have:
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