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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