IMPLICIT DIFFERENTIATION

[racuna]

The problem is
8+5x=sin(xy^4)

I have to take the derivative of both sides with respect to x, right?
This is what I got
5=cos(4xy^3y'+y^4)

I'm lost from there.

 

[stapel]

I have to take the derivative of both sides with respect to x, right?

If you were asked to find dy/dx or y', then, yes.

Apply the Chain Rule to the right-hand side. The derivative of sin(y) is not cos(dy/dx), but [cos(y)][dy/dx]. So differentiate the sine, leaving the argument untouched, and then multiply by the derivative of that argument.

To complete the solution, if the instructions were to "find dy/dx" or "isolate y'" or something similar, you will need to multiply out the [dy/dx] part (which will be a sum) against the [cos(y)] part. Subtract the "y⁴cos(xy⁴)" term to the left-hand side, and divide off the "4xy³" to isolate "dy/dx" on the right-hand side.

Hope that helps a bit. If you get stuck, please reply showing how far you have gotten. Thank you.

Eliz.

 

[Unco]

You've got the right idea, but your use of the chain rule needs some work.

An easier example of the chain rule:

f(x) = sin(2x)

Differentiate the 'outer', sin: cos

This acts on the normal 'inner': cos(2x)

Multiply by the derivative of the 'inner': cos(2x) * 2

Your implicit differentiation is spot on (you've just put in the wrong place).

Don't forget to rearrange it to get y' =...

 

[racuna]

what is the y^4cos(xy^4) part? Is this the chain rule?

 

[Unco]

Implicitly differentiate sin(xy^4) with the chain rule first, and you will be equipped to to understand what Eliz was saying there.

 

[racuna]

ok

Thanks. I think I get that part now.