suppose f(x)+ x^4[f(x)]^3=1028 and f(2)=4, find f`(2) How do I start this?

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suppose f(x)+ x^4[f(x)]^3=1028 and f(2)=4, find f`(2) How do I start this?

Mathematics
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YAY differentiation!
F(x) + x^4 [f(x)]^3 = 1028
let me replace f(x) by y so that its easier to type

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y + x^4*y^3 = 1028; Use implicit differentiation: dy/dx + (x^4 * 3y^2 *dy/dx + 4x^3 * y^3) = 0
I used the chain rule to differentiate that x^4*y^3
So now just replace dy/dx by f'(x) and y by f(x), and then find the values at 2: f'(2) + (2^4 * 3 * [f(2)]^2 * f'(2) + 4*2^3 * [f(2)]^3) = 0
Replace f(2) by 4, because f(2) = 4 and simplify a bit: f ' (2) + (16 * 3 * 4^2 * f ' (2) + 4 * 8 * 4^3) = 0 f ' ( 2 ) + 768 * f ' ( 2 ) + 2048 = 0
wait sorry i made a typo
checkin my arithmetic... i hate typing out math solutions
okay so here we go: \[y + x^4 * y^3 = 1028\] \[dy/dx + x^4 * 3y^2 (dy/dx) + y^3 *4x^3 = 0\] \[dy/dx( 1 + x^4 * 3y^2 ) = -y^3 * 4x^3\]
\[dy/dx = (-y^3*4x^3)/(1+x^4*3y^2)\]
okay now you know that f(2) is 4, therefore to find f' or dy/dx at x = 2, replace all ys by 4 and xs by 2
\[dy/dx = (-4^3 * 4*2^3) / (1 + 2^4 * 3* 4^2)\]
dy/dx = (-2048)/769
dy/dx = -2048/769
I don't like the number.. don't know why, but i triple checked my math...
WOOOT! Thanks for fanning, whoever you are, lol I'm a superhero now!

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