If $f(x) + x^2[f(x)]^3 = 10$ and $f(1) = 2$, find $f '(1)$

Question

If $f(x) + x^2[f(x)]^3 = 10$ and $f(1) = 2$, find $f '(1)$.

I'm not entirely sure how to solve this problem. My first instinct would be to plug in a 2 for every $f(x)$ and then take the derivative, but that just gives me zero. Could someone please explain this to me?

Answer 1

Hint: try differentiating the equation (implicitly). If this does not seem very obvious to you, try substituting $y=f(x)$ and then try again.

Answer 2

Start with $$ f(x) + x^2 f(x)^3 = 10 $$ Take derivatives of both sides $$ f'(x) + 2x f(x)^3 + x^2 3f(x)^2 f'(x) = 0. $$ Set $x = 1$ in the last equation $$ f'(1) + 2f(1)^3 + 3f(1)^2 f'(1) = 0 $$ As $f(1) = 2$, we have $$ f'(1) + 16 + 12 f'(1) = 0 \iff f'(1) = -\frac{16}{13} $$

Answer 3

If one assumes $f$ is differentiable near $x=1$ and if one assumes the first equality is an identity, then one may differentiate it to get $$ f'(x)+2x \cdot [f(x)]^3+x^2 \cdot 3 \times f'(x)\cdot [f(x)]^2=0 $$ then putting $x:=1$ gives $$ 13f'(1)=-16. $$