Please help with implicit differentiation. Find dy/dx for 4x^3+x^2y-xy^3 = -4
4x^3+x^2y-xy^3 = -4
we want derivatives with respect to x, so whenever we get a dx/dx, we get 1
So I will not write the dx/dx 's
12x^2 + (x^2)dy/dx + y(2x) - x(3y^2)dy/dx - y^3 = 0
dy/dx(x^2 - 3xy^2) = y^3 - 12x^2
dy/dx = (y^3 - 12x^2)/(x^2 - 3xy^2)
check my work, I tend to make errors when I don't do it on paper first
notice I needed to use the product rule for the 2nd and 3rd terms
Hi Reiny. According to my webwork. The answer it not correct. Any additional help?
sorry, I forgot the +2xy on the left side
should have been
dy/dx = (y^3 - 2xy - 12x^2)/(x^2 - 3xy^2)
I am disappointed that you could not find that simple error.
this problem has been a challenge for me.
To find dy/dx using implicit differentiation, follow these steps:
1. Start by differentiating both sides of the equation with respect to x. Treat y as a variable with respect to x, so all terms involving y will be differentiated using the chain rule.
2. When differentiating x^2y, we treat it as a product of two functions: x^2 and y. Using the product rule, we differentiate x^2 with respect to x (which is 2x) and multiply it by y. Then, differentiate y with respect to x.
3. Similarly, when differentiating xy^3, we treat it as a product of two functions: x and y^3. Using the product rule, we differentiate x with respect to x (which is 1) and multiply it by y^3. Then, differentiate y^3 with respect to x using the chain rule (which is 3y^2(dy/dx)).
4. After differentiating both sides, collect the terms that involve dy/dx on one side. All other terms, including constants, should be collected on the other side.
Now, let's apply these steps to solve the given equation: 4x^3 + x^2y - xy^3 = -4.
Differentiating both sides:
12x^2 + 2xy + x^2(dy/dx) - y^3 - 3xy^2(dy/dx) = 0
Collecting the terms involving dy/dx on one side:
x^2(dy/dx) - 3xy^2(dy/dx) = -12x^2 - 2xy + y^3
Factor out dy/dx:
(dy/dx)(x^2 - 3xy^2) = -12x^2 - 2xy + y^3
Divide both sides by (x^2 - 3xy^2):
dy/dx = (-12x^2 - 2xy + y^3) / (x^2 - 3xy^2)
That is the derivative dy/dx for the equation 4x^3 + x^2y - xy^3 = -4.