Mastering the Chain Rule: A Comprehensive Guide to Differentiating Composite Functions in Calculus

Chain Rule

The chain rule is a fundamental rule in calculus that allows us to differentiate composite functions

The chain rule is a fundamental rule in calculus that allows us to differentiate composite functions. It is used when we have a function within another function.

Let’s say we have two functions: f(x) and g(x). The chain rule tells us how to differentiate the composite function f(g(x)). The general form of the chain rule is given by:

d/dx [f(g(x))] = f'(g(x)) * g'(x)

Here, f'(g(x)) represents the derivative of the outer function f with respect to the inner function g(x), and g'(x) represents the derivative of the inner function g(x) with respect to x.

To apply the chain rule, follow these steps:

Step 1: Identify the composite function. Let’s call this composite function y.

Step 2: Identify the inner function and the outer function within y.

Step 3: Differentiate the outer function with respect to the inner function.

Step 4: Differentiate the inner function with respect to x.

Step 5: Multiply the derivatives obtained in Steps 3 and 4.

Step 6: Simplify the result if necessary.

Let’s go through an example to illustrate the chain rule:

Example:
Find the derivative of the function y = (2x^3 + 5x – 1)^4.

Step 1: Identify the composite function y = (2x^3 + 5x – 1)^4.

Step 2: The inner function is 2x^3 + 5x – 1, and the outer function is y = u^4, where u = 2x^3 + 5x – 1.

Step 3: Differentiate the outer function with respect to the inner function.
By using the power rule, we get: dy/du = 4u^3.

Step 4: Differentiate the inner function with respect to x.
We differentiate each term of the inner function separately:
d/dx (2x^3) = 6x^2,
d/dx (5x) = 5,
d/dx (-1) = 0.
So, du/dx = 6x^2 + 5.

Step 5: Multiply the derivatives obtained in Steps 3 and 4.
dy/dx = (dy/du) * (du/dx) = 4u^3 * (6x^2 + 5).

Step 6: Simplify the result if necessary.
Substitute u = 2x^3 + 5x – 1 back into the expression for dy/dx:
dy/dx = 4(2x^3 + 5x – 1)^3 * (6x^2 + 5).

Therefore, the derivative of y = (2x^3 + 5x – 1)^4 is dy/dx = 4(2x^3 + 5x – 1)^3 * (6x^2 + 5).

The chain rule is a powerful tool that allows us to differentiate more complicated functions by breaking them down into simpler parts. Practice applying the chain rule with different examples to solidify your understanding.

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