Implicit Differentiation

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October 1st 2012, 08:17 PM
phenol

Implicit Differentiation

Hello,
Would anyone be able to tell me how to go about solving this problem? I've been looking through my calculus (it doesn't give examples similar to this problem) book but I have no idea where to start. Any help would be greatly appreciated. For the 1st part, I got dy/dx = -(10x+3+y)/x. Not sure if it's correct or not. Am I supposed to plug 2 in to find the slope? Thanks! Ana

Here's the problem . . .
5x²+ 3x + xy = 2 and y(2) = -12, find y'(2) by implicit differentiation. (1st part)

2nd part asks to find the equation of the tangent line to the graph at the point (2,-12)
October 1st 2012, 08:38 PM
MarkFL

Re: Implicit Differentiation

You differentiated correctly, now use the given point (2,-12) to find y'(2). Then use the point-slope formula to find the equation of the tangent line.
October 1st 2012, 09:16 PM
AZach

Re: Implicit Differentiation

Instead of making a new thread on these, I want to just pop in a quick question to see if I implicitly differentiated this correctly; problem is $(6x-y)^4+3y^3=9976$ , and I implicitly differentiated y with respect to x to get this $dy/dx = \frac{(24(6x-y)^3)}{(1+9y^2)}$ ... Does that look about right? I have a gut feeling it isn't, but I've combed through it a few times and I'm not sure how else to differentiate it. Thanks!
October 1st 2012, 09:48 PM
MarkFL

Re: Implicit Differentiation

I get a different result...can you post your work?
October 1st 2012, 09:53 PM
phenol

Re: Implicit Differentiation

Thanks for your help, I think I got it. The slope came out to -11/2 and the equation for the tangent line came out to be y=-11/2(x)-1. Thanks again!!
October 1st 2012, 09:57 PM
MarkFL

Re: Implicit Differentiation

Yes, I get the same result. :)
October 2nd 2012, 11:03 AM
AZach

Re: Implicit Differentiation

Quote:

Originally Posted by MarkFL2

I get a different result...can you post your work?

Should've posted it originally, my bad. Here it is:

$(6x-y)^4+3y^3=9976$

$d/dx[(6x-y)^4]+d/dx[3y^3]=d/dx[9976]$ (Took derivative with respect to x)

$4(6x-y)^3*6*(-1)*(-dy/dx)+9y^2*dy/dx=0$

$-24(6x-y)^3*(-dy/dx)+9y^2*dy/dx=0$ (multiplied 6 and -1 by 4)

$dy/dx(24(6x-y)^3+9y^2)=0$ (factored out a dy/dx and got rid of the -1 by multiplying it by -24)

This is where I'm confused, and I have a feeling it's because I'm not doing something right when I differentiated the inside function -y from (6x-y)^4?

Or I should revisit step 4 and divide the left side and right side of the equation by -24.

$\frac{-24(6x-y)^3*(-dy/dx)+9y^2*dy/dx}{-24}=\frac{0}{-24}$

$dy/dx[(6x-y)^3*-1+9y^2]=0$ (now factor out dy/dx)

Actually that's still going to be wrong because dy/dx isn't going to be zero. I must need to add or subtract something from the left to get a quantity on the right?