Math Tutor

The chain rule is a fundamental concept in calculus that allows us to differentiate composite functions. In other words, it allows us to find the derivative of a function that is composed of two or more functions. The chain rule states that if we have a composite function f(g(x)), where g(x) is a differentiable function and f(x) is a differentiable function, then the derivative of f(g(x)) with respect to x is given by: f'(g(x)) * g'(x) This can be understood as finding the derivative of the outer function f(x) and multiplying it by the derivative of the inner function g(x). Here's an example to help illustrate this concept: Let f(x) = x^2 and g(x) = 2x. We want to find the derivative of the composite function h(x) = f(g(x)) = f(2x) = (2x)^2. Using the chain rule, we have: h'(x) = f'(g(x)) * g'(x) = 2 * (2x) * (2) = 8x So the derivative of h(x) = 8x. Another example: let f(x) = e^x and g(x) = sin(x). We want to find the derivative of h(x) = f(g(x)) = f(sin(x)) = e^(sin(x)). Using the chain rule, we have: h'(x) = f'(g(x)) * g'(x) = e^(sin(x)) * cos(x) So the derivative of h(x) = e^(sin(x)) * cos(x). I hope this helps! Let me know if you have any questions or would like to see additional examples.