/ 2 \ log\x + 2*x/
d / / 2 \\ --\log\x + 2*x// dx
Let .
The derivative of is .
Then, apply the chain rule. Multiply by :
Differentiate term by term:
Apply the power rule: goes to
The derivative of a constant times a function is the constant times the derivative of the function.
Apply the power rule: goes to
So, the result is:
The result is:
The result of the chain rule is:
Now simplify:
The answer is:
/ 2\ | 2*(1 + x) | 2*|1 - ----------| \ x*(2 + x) / ------------------ x*(2 + x)
/ 2\ | 4*(1 + x) | 4*(1 + x)*|-3 + ----------| \ x*(2 + x) / --------------------------- 2 2 x *(2 + x)