The chain rule is a powerful tool used in the process of differentiation, especially for composite functions. Composite functions combine two or more functions, resulting in a new function known as a function of a function.
The formula for the chain rule is as follows:
Chain rule formula:
if y is a function of u and u is a function of x, then dy/dx = dy/du * du/dx.
Function notation:
if f is a function of x, then f'(x) = f'(u) * du/dx.
To better understand the chain rule, let's examine some examples:
By applying the chain rule formula, we can easily solve for dy/dx in each example.
Sometimes, the given function may be in function notation and we are asked to differentiate it. In this case, the steps are similar:
If the function given is not in the form y = f(u), we can use a different formula:
Reverse chain rule formula:
if u = f(x), then f'(x) = dy/du * du/dx.
For example:
Example 3:
Find the value of dy/dx at the point (3,5) on the curve x = 2y - y².
We can easily solve for dy/dx using the reverse chain rule formula by differentiating with respect to y and substituting the values into the formula.
The chain rule is also useful in the integration process, known as the reverse chain rule. The steps are:
Understanding the chain rule allows for easier handling of complex functions. By applying the correct formula, you can effortlessly differentiate and integrate composite functions with confidence.