When you have a function inside a function, differentiate the outside, then multiply by the derivative of the inside.
Walk through
Step 1 of 4
Spot the inside and outside
For y = sin(x²): the outside function is sin(·), the inside is g(x) = x².
Chain rule
Differentiate the outer layer, then multiply by the derivative of what's inside. Layer by layer.
Your turn
Differentiate y = (3x + 1)⁵.
Watch out
Forgetting to multiply by the inside derivative. d/dx[cos(2x)] is −2sin(2x), not just −sin(2x).
Chain rule
Try it
d/dx(sin(x²))
cos(x²) · 2x.