What are some advanced examples of the chain rule in calculus?

What are some advanced examples of the chain rule in calculus?

The chain rule is a fundamental concept in calculus used to differentiate composite functions. An advanced example is differentiating a function like sin(x^2). Would you like to see the step-by-step process?

Yes, please show me the steps for differentiating sin(x^2).

Sure! To differentiate sin(x^2), we apply the chain rule. First, let u = x^2, then the derivative of sin(u) is cos(u) * du/dx. So, we have cos(x^2) * 2x. Thus, the derivative is 2x * cos(x^2). Do you want to explore more examples?

Can you give me another example involving exponential functions?

Absolutely! Consider the function e^(3x^2). Using the chain rule, we let u = 3x^2. The derivative is e^(u) * du/dx = e^(3x^2) * 6x. Would you like to dive deeper into any specific applications?