What is the chain rule in calculus?

What is the chain rule in calculus?

The chain rule is a formula for computing the derivative of a composite function. It states that if you have two functions, f(g(x)), the derivative is f'(g(x)) * g'(x). Would you like a visual representation of this?

Yes, can you show me a graph that illustrates the chain rule?

Sure! Imagine you have two functions: f(x) = x^2 and g(x) = sin(x). The chain rule helps us find the derivative of their composition. I can create a graph showing how the slopes of these functions relate.

That sounds great! How do I interpret the graph?

In the graph, the slope of f(g(x)) at any point is the product of the slope of f at g(x) and the slope of g at x. This shows how changes in x affect the overall function.

Can you give me an example with actual numbers?

Absolutely! Let's take g(x) = x + 1 and f(x) = x^2. If x = 2, then g(2) = 3 and f(g(2)) = f(3) = 9. The derivative at that point would be f'(3) * g'(2). Would you like to calculate that?