Question

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

Answer 1

YAY differentiation!

Answer 2

F(x) + x^4 [f(x)]^3 = 1028

Answer 3

let me replace f(x) by y so that its easier to type

Answer 4

y + x^4*y^3 = 1028;
Use implicit differentiation:
dy/dx + (x^4 * 3y^2 *dy/dx + 4x^3 * y^3) = 0

Answer 5

I used the chain rule to differentiate that x^4*y^3

Answer 6

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

Answer 7

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

Answer 8

wait sorry i made a typo

Answer 9

checkin my arithmetic... i hate typing out math solutions

Answer 10

okay so here we go:
\[y + x^4 * y^3 = 1028\]
\[dy/dx + x^4 * 3y^2 (dy/dx) + y^3 *4x^3 = 0\]
\[dy/dx( 1 + x^4 * 3y^2 ) = -y^3 * 4x^3\]

Answer 11

\[dy/dx = (-y^3*4x^3)/(1+x^4*3y^2)\]

Answer 12

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

Answer 13

\[dy/dx = (-4^3 * 4*2^3) / (1 + 2^4 * 3* 4^2)\]

Answer 14

dy/dx = (-2048)/769

Answer 15

dy/dx = -2048/769

Answer 16

I don't like the number.. don't know why, but i triple checked my math...

Answer 17

WOOOT! Thanks for fanning, whoever you are, lol I'm a superhero now!