Chain Rule For Finding Derivatives


To solve the derivative of function f which is f(x) = (2x + 3)2, the way you have used so far is to describe the form first into the sum of terms, where the result is f(x) = 4x2 + 12x + 9. What if we are asked to solve a derivative of a function such as f(x) = (x3 + x2 -4x + 5)5 ?. Even though we can describe such a form, it is clearly not effective. So to solve such derivative multiple functions can be used a method called the Chain Rule.
A compound function that we can see as the composition of the function f o g = f [g(x)] is interpreted as "the function f continues the function g" for g(x) = x3 + x2 -4x + 5  and f(x) = x5 as presented in the arrow diagram (Figure 1) below,




Example 1.
If f(x) = (x2 + 3x)2 is a function composition f o g, then specify f and g.
Answer:
f(x) = x2 and g(x) = x2 + 3x

Example 2.
If f(x) = cos3 (3x -4) is a function composition f o g, then specify f and g.
Answer:
f(x) = x3 and g(x) = cos(3x -4).

Chain Theorem


If we have gotten a picture of how to decipher compound functions into two single functions, the question is how can we obtain a rule for finding the derivative of that function. For that, consider the following steps.

Let f(x) = f[g(x)], g(a) = b, and g(a + h) = b + k.



For g(a + h) = b + k = g(a) + k, then k = g(a + h) - g(a), so


For k ==> 0, then


F’(x) = f’ [g(x)] g’(x)

These formulas are called chain theorems. In practice, chain theorems are often expressed by the following Leibniz Notation,

If f and g are differentiable functions and the composition of functions defined by y = f [g (x)] is also differentiable, then the derivative of the function's composition can be expressed by the formula:


Example 3.
Find the derivative of function y = F(x) = (3x - 5)4.
Answer:
For example, y = u4 and u = 3x - 5, then


Example 4.
Find the derivative of function 

For example, y = u-1 and u = 4x – x2, then





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