Derivative question..(x+y)^12 + (x-y)^12 =a ; find for d2y/dx..?

Question

how can i solve this problem...what is the answer and solution..please help me..

Answer 1

(x + y)^12 + (x - y)^12 = a

And you want to solve for the second derivative, d^2y/dx^2.

First, we differentiate implicitly.

12(x + y)^11 [1 + dy/dx] + 12(x - y)^11 [1 - dy/dx] = 0

Expand by distributing to the 1 and to the dy/dx. (Do NOT expand
(x + y)^11 or (x - y)^11).

12(x + y)^11 + (dy/dx)12(x + y)^11 + 12(x - y)^11 -
(dy/dx) 12(x - y)^11 = 0

Now, bring everything that doesn't have a dy/dx over to the right hand side.

(dy/dx)12(x + y)^11 - (dy/dx) 12(x - y)^11 = -12(x + y)^11 - 12(x - y)^11

Let's factor out 12 dy/dx on the left hand side, and -12 on the right hand side.

12(dy/dx) [(x + y)^11 - (x - y)^11] = -12[(x + y)^11 + (x - y)^11]

Now, divide both sides to isolate dy/dx. Note that the 12 will cancel with the -12, giving us

(dy/dx) = -[(x + y)^11 + (x - y)^11] / [(x + y)^11 - (x - y)^11]

Let's apply the minus sign on the outside of this fraction to the denominator. This would reverse the order.

(dy/dx) = [(x + y)^11 + (x - y)^11] / [(x - y)^11 - (x + y)^11]

At this point, we solve for the second derivative. I'll leave it up to you to complete, but you pretty much solve this implicitly as well.

Answer 2

Use implicit differentiation: take d/dx of all the terms. Do this twice.

First time get 12*(x+y)^11*(1+dy/dx) + 12*(x-y)^11*(1-dy/dx) = 0;

next differentiation gives

11*12*(x+y)^10*(1+dy/dx)^2 + 12(x+y)^11*d^2y/dx^2 + 11*12*(x-y)^10*(1-dy/dx)^2+ 12*(x-y)^11*(-d^2y/dx^2)

Solve the second equation for d^2y/dx^2 in terms of x, y and dy/dx.
Solve the first for dy/dx and substitute for that.


It's messy but straightforward.

Answer 3

Differentiate both sides of the equation with respect to x. The right hand side of course goes to zero. Do it again.

With any luck, you will find that the terms on the right involving dy/dx will cancel to zero. What's left should let you solve for the second derivative of y with respect to x.

I think that will work out nicely, but I haven't checked it carefully.

Answer 4

dy/dx=12(x+y)^11+12(x-y)^11=0
d2y/dxdx=11*12(x+y)^10+11*12(x-y)^10=0