Question

Find dy/ dx at x = 2 if y = 6 (u square 2) - 11 and u = 3 (x square 2) + 2

Answer 1

There are two ways to do this problem, but usually problems like this are geared to teach you how to use the chain rule. So thats what we will do.
First, take the derivative of y with respect to x (not the middle variable u). This gives:
\[y = 6u^2-11 \Rightarrow y' = 12u*u'\]
that one part (12u*u') is the chain rule in effect.
The second step is to now take the derivative of u with respect to x:
\[u = 3x^2+2 \Rightarrow u' = 6x\]
The last step is just to plug the values of u' and u into the equation for y':
\[y ' = 12u*u' \Rightarrow y' = 12(3x^2+2)*(6x) = 216x^3+144x\]
if x = 2, we get:
y' = 216(8)+144(2) = 2016