Finding the Derivative of a Function Given an Equation and Initial Condition

[momogiri]

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

Attempt:
I wish I could say I tried, but I don't know how to approach this problem..
All I did was double check the formula worked by inputting 2 for f(x) and 1 for x

Can someone tell me how to start this? And I'll go from there :)

 

[ZioX]

Take the derivative of both sides of the equation.

 

[NonAbelian]

And don't forget to use the chain rule on the second part. You'll end up with an equation that involves both f(x) and f'(x). At this point, sub in x=1 and f(1)=2 and solve for f'(x).

 

[momogiri]

Ok, so what I've done is:
take derivative of both sides
original = $$f(x) + x^{2}(f(x))^{3} = 10$$
so d/dx of f(x) is f'(x)
now $$x^{2}(f(x))^{3}$$'s derivative I'm a bit unsure of..

What I did was use the product rule, so..
$$x^{2}((f(x))^{3})' + (f(x))^{3}(x^{2})'$$
and $$(f(x)^{3})' = 3(f(x))^{2}*f'(x)$$ right?
So...
it's $$x^{2}(3(f(x))^{2}*f'(x)) + f(x)^{3}(2x)$$
Then that means the whole equation becomes
$$f'(x) + x^{2}(3(f(x))^{2}*f'(x)) + f(x)^{3}(2x) = 0$$
In which I plugged in the numbers
so..
$$f'(1) + (1)^{2}(3(2)^{2}*f'(1)) + (2)^{3}(2(1)) = 0$$
then
$$f'(1) + 12*f'(1)) + 16 = 0$$
which means
$$(13)f'(1)) = -16$$
making
$$f'(1) = -16/13$$?
Does that make sense? Is it wrong? i have a feeling it is :/

 

[bob1182006]

looks right.

if you wanted to not deal with f(x) you could change f(x)=y and do implicit differentiation like you've been doing before and solve for y' and then change y to f(x).

 

[eyehategod]

f(x)+x^(2)*f(x)^(3)//use product rule for the second term along with the chain rule
f'(x)+x^(2)*3f(x)^(2)f'(x)+2xf(x)^(3)
f'(x)[1+3x^(2)]=-2xf(x)^(3)//moved 2xf(x)^(3) to other side and factored a f'(x)
f'(x)=[-2xf(x)^(3)]/[1+3x^(2)]
f'(1)=[-2(1)f(1)^3]/[1+3(1)^(2)}
f'(1)=-16/13
OWNED!