The chain rule is a fundamental tool in calculus used to differentiate composite functions. It allows us to find the derivative of a function that is composed of two or more functions. Essentially, if you have a function nested inside another function, the chain rule helps us to understand how changes in one variable affect another.
Here's how the chain rule works in a simplified manner:
- Identify the "outer" and "inner" functions.
- Differentiate the "outer" function with respect to the "inner" function.
- Differentiate the "inner" function with respect to the variable of interest.
- Multiply these derivatives together.
For example, in the function given in the problem (a) "2sin5θ", the outer function is "2sin(u)" and the inner function is "u = 5θ". By applying the chain rule, you first find the derivative of the sine function, then multiply it by the derivative of the inner function, 5θ, giving the result.
The chain rule is crucial for differentiating compositions of functions, making it invaluable in more advanced calculus applications.