If $f(x^3 + x) = x^3 + x^2 + 1$, then what is $f'(2)$?

Question

I don't even have an idea of how to solve this problem. I solved every single problem in my text book until this question so I thought I'm either missing some critical information about derivatives or just couldn't find the way to solve.

Thanks in advance.

score 15 · Accepted Answer · 2016-01-22T18:05:46.287

15

Use the chain rule :

$$g(x)=x^3+x^2+1=f(x^3+x)$$ so : $$3x^2+2x=f'(x^3+x)(3x^2+1)$$ Now just let $x=1$ to get $$5=4f'(2)$$ $$f'(2)=\frac{5}{4}$$

edited Jan 22 '16 at 18:05

answered Dec 14 '15 at 18:46

score 8 · Answer 2 · answered Dec 14 '15 at 18:44

8

Well, if you take a derivative of the LHS you get $$ f'(x^3+x) (3x^2+1), $$ so you can solve for $f'(x^3+x)$ and pick $x$ so that $x^3+x=2$

answered Dec 14 '15 at 18:44

gt6989b

54,422

If $f(x^3 + x) = x^3 + x^2 + 1$, then what is $f'(2)$?

2 Answers2