# 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) = 4x^{2}+ 12x + 9. What if we are asked to solve a derivative of a function such as f(x) = (x^{3}+ x^{2}-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) = x

^{3}+ x^{2}-4x + 5 and f(x) = x^{5}as presented in the arrow diagram (Figure 1) below,**Example 1**.

If f(x) =
(x

^{2}+ 3x)^{2}is a function composition f o g, then specify f and g.
Answer:

f(x) = x

^{2}and g(x) = x^{2}+ 3x**Example 2**.

If f(x) =
cos

^{3 }(3x -4) is a function composition f o g, then specify f and g.
Answer:

f(x) = x

^{3}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 = u

^{4}and u = 3x - 5, then**Example 4**.

Find the derivative of function

For example, y = u

^{-1}and u = 4x – x^{2}, thenSUBSCRIBE TO OUR NEWSLETTER

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