Derivadas de funciones compuestas

What is the chain rule?

In mathematics, the chain rule is essential for the differentiation of composite functions. This concept is crucial, since most functions apply this rule. Basically, it allows you to derive complex functions by breaking them down into external and internal functions. First, the external function is derived without altering the internal function, and then it is multiplied by the derivative of the internal function. This sequential process justifies the term "chain". Let us see how it is applied in different contexts.

How to derive a composite function with a constant multiple?

To understand the procedure, let us consider an example. Suppose we have a function f(x) = -√(x + 2). Here, the constant multiple is -1. We decompose the function into its outer part, the square root, and the inner part, a linear function. We proceed to:

1. Derive the outer function without touching the inner one:

   h(x) = √x = x^(1/2)h'(x) = (1/2)x^(-1/2) = 1 / (2√x).

2. Substitute x in the derivative of h(x) for the inner function:

   1 / (2√(x + 2))

3. Multiply by the derivative of the inner function (x + 2):

   The derivative is 1, resulting in the original function:

   f'(x) = -1 * 1 / (2√(x + 2))

How to handle functions with three levels of composition?

A more complex case arises when we have functions with several levels such as h(f(g(x))). The chain rule is applied successively to each level:

1. Derive the outer function, without altering the inner levels.
2. Continue deriving each inner function at the next level.

This illustrates the versatility of the chain, applicable even to complex compositions, thus ensuring correct differentiation in advanced calculations such as in engineering and business.

Example with power functions

Let's see an example applying this knowledge to a function with a constant multiple:

f(x) = (1/7) * (1 - x)^2.

1. The outer function is the power, and the inner function is the operation inside the parenthesis. We derive:

   h(x) = x^2h'(x) = 2x.

2. Applying the chain rule, we substitute:

   f'(x) = (1/7) * (2 * (1 - x)^1) * -1.

   This translates to:

   f'(x) = - (2/7) * (1 - x)

Derivation of trigonometric functions

This technique also applies to trigonometric functions. For example, given f(x) = sin(2x):

1. The derivative of sine is cosine:

   f'(x) = cos(2x) * d(2x)/dx.

2. We calculate the derivative of the internal derivative, which is 2. Thus:

   f'(x) = 2 * cos(2x).

What about exponential functions?

When dealing with functions like e^(√x) / 3:

1. The outer function is e and the inner function is √x. We apply the same technique:

   f'(x) = (1/3) * e^(√x) * d(√x)/dx.

2. We derive the square root function:

   1 / (2√x)

3. The final result is:

   f'(x) = (1/6√x) * e^(√x).

The importance of practice in the chain rule.

The complexity of this rule emphasizes the importance of practice. Several exercises mix various rules, so patience and dedication are essential. It is always helpful to share with the community and exercise everything you have learned before looking for answers. Remember, the key is to keep practicing and to rely on the community if any doubt arises. Cheer up!