How do I apply the chain rule to sin(x^2+1) with three layers?

[Dennis_T]

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[Richard46]

I’m really struggling to understand how to apply the chain rule when the function is something like sin(x² + 1). I get the basic idea of outside and inside derivatives, but when I try to differentiate something more layered, like a composite of three functions, my work gets messy and I’m never sure if I’m simplifying correctly.

[PatrickL]

I used to freeze when there are three layers. I would scribble and then realize I must take the inner function first and then go outward. It helped a bit but I still trip on the last step.

[Eleanor_T]

For sine of x squared plus one the outer function is sine and its derivative is cosine. Then multiply by the derivative of the inside which is two times x. If I set u to the inside I see du by dx equals two x. Still feels clumsy on a fast problem.

[MatthewT]

I tried with three functions once and my notes looked like a maze. I kept losing track of what is what and I ended up with a wrong result maybe. I would draw arrows and label inner and outer but I still slip.

[Kyle93]

Maybe the real problem is not the rule but the bookkeeping. naming the inner function and writing the derivative of that piece can be where things go wrong.

[Kevin_L]

A tutor told me to write the chain rule as follow the outer function then the next inner function then the last inner. I tried and it helped a little but my handwriting was rushed and I skipped a step.

[Kyle52]

If you have a problem like sine of x squared plus one I think the derivative is cosine of the inner thing times the derivative of the inner thing. So cosine of the inner thing times the derivative of the inner thing and it can still feel hazy.

[Anthony98]

Sometimes I drift off during a problem like this and wonder if maybe the real issue is not the chain rule but how I keep track of the layers. Then I snap back and try to slow down, but the scramble returns.