Use implicit differentiation to find dy/dx if cos xy = 2x^2 - 3y.
I'm stuck on this problem because I'm getting thrown off on how to factor this. Here's my work so far:
-{[(dy/dx)y + y]sin xy} = 4x - 3(dy/dx)
-{[(dy/dx)y + y]sinxy} + 3(dy/dx) = 4x
...now what?
Thx
factor out dy/dx which is what you want
dy/dx [ 3 -y sinxy] = [ 4 x + y sin xy ]
so
dy/dx =[ 4 x + y sin xy ] / [ 3 -y sinxy]
There is a minor correction to the original differentiation:
-{[(dy/dx)x + y]sin xy} = 4x - 3(dy/dx)
To solve this problem using implicit differentiation, we will differentiate both sides of the equation with respect to x, treating y as a function of x. Let's go step by step:
1. Start with the equation cos(xy) = 2x^2 - 3y.
2. Take the derivative of both sides with respect to x. Keep in mind that y is a function of x, so we need to use the chain rule:
d/dx(cos(xy)) = d/dx(2x^2 - 3y)
3. On the left side, we have to use the chain rule. Let's break it down:
d/dx(cos(xy)) = -(dy/dx)(y)sin(xy)
Notice that we have the product of three functions: -(dy/dx), y, and sin(xy). To differentiate this product, we use the product rule. The derivative of the first function -(dy/dx) is -(d^2y/dx^2), and the derivative of the second function y is dy/dx. Finally, the derivative of the last function sin(xy) is cos(xy) times the derivative of the inside, which is (dy/dx)xy + yx.
Putting it all together:
-(dy/dx)(y)sin(xy) = -(d^2y/dx^2)y - 3(dy/dx)xy - 3y
4. On the right side, we differentiate each term:
d/dx(2x^2) = 4x
d/dx(-3y) = -3(dy/dx)
5. Now rewrite the equation with the derivatives:
-(d^2y/dx^2)y - 3(dy/dx)xy - 3y = 4x - 3(dy/dx)
6. Rearrange the equation to solve for dy/dx:
-(d^2y/dx^2)y - 3(dy/dx)xy + 3(dy/dx) = 4x + 3y
7. Factor out dy/dx:
[3 - 3xy] (dy/dx) = 4x + 3y + (d^2y/dx^2)y
8. Finally, solve for dy/dx by dividing both sides by (3 - 3xy):
dy/dx = (4x + 3y + (d^2y/dx^2)y) / (3 - 3xy)
That is the implicit derivative dy/dx in terms of x and y. Keep in mind that this problem can involve further simplification or calculations depending on the specific values of x and y.