How to Differentiate the Function of a Function?

Differentiating a function of a function, also known as a composite function, involves using the chain rule.

The chain rule provides a method to find the derivative of a composite function, which is a function that is composed of another function. In simpler terms, it deals with situations where you have a function inside another function. The chain rule is one of the fundamental rules of calculus.

The Chain Rule Explained

If you have a function y = f(g(x)), where f and g are differentiable functions, then the derivative of y with respect to x is given by:

dy/dx = f'(g(x)) * g'(x)

In words:

• Take the derivative of the outer function f with respect to its argument, leaving the inner function g(x) intact. This gives you f'(g(x)).
• Multiply that result by the derivative of the inner function g(x) with respect to x. This gives you g'(x).

Step-by-Step Breakdown

Here's a step-by-step approach to applying the chain rule:

1. Identify the outer and inner functions. Look for a function "inside" another function.
2. Find the derivative of the outer function. Remember to keep the inner function as its argument.
3. Find the derivative of the inner function.
4. Multiply the two derivatives together. This gives you the derivative of the composite function.

Examples

Let's illustrate with a few examples:

• Example 1: y = sin(x²)

  • Outer function: f(u) = sin(u)

  • Inner function: g(x) = x<sup>2</sup>

  • f'(u) = cos(u)

  • g'(x) = 2x

  • Applying the chain rule: dy/dx = cos(x<sup>2</sup>) * 2x = 2x * cos(x<sup>2</sup>)

• Example 2: y = (3x + 1)⁵

  • Outer function: f(u) = u<sup>5</sup>

  • Inner function: g(x) = 3x + 1

  • f'(u) = 5u<sup>4</sup>

  • g'(x) = 3

  • Applying the chain rule: dy/dx = 5(3x + 1)<sup>4</sup> * 3 = 15(3x + 1)<sup>4</sup>

• Example 3: y = e^-x

  • Outer function: f(u) = e<sup>u</sup>

  • Inner function: g(x) = -x

  • f'(u) = e<sup>u</sup>

  • g'(x) = -1

  • Applying the chain rule: dy/dx = e<sup>-x</sup> * (-1) = -e<sup>-x</sup>

Tips and Considerations

• Practice is key. The more you practice applying the chain rule, the more comfortable you'll become.
• Be careful with notation. Pay close attention to the order of operations and the arguments of the functions.
• The chain rule can be applied multiple times. If you have a function composed of multiple functions, you'll need to apply the chain rule repeatedly.

By understanding and applying the chain rule, you can differentiate complex functions that involve function composition effectively. Remember to identify the outer and inner functions, differentiate each separately, and then multiply the results.