How to find dy/dx from x^2y+3xy^2-x=5?

2 Answers
Mar 9, 2018

dy/dx=(-2xy-3y^2+1)/(x^2+6xy)

Explanation:

To solve this, we need to use implicit differentiation.

Since we are finding the derivative in respect to x, anytime we find the derivative of y, we need to tag on a dy/dx after.

To start off, we need to use the Product Rule for the first two terms, since there are two variables being multiplied together.

The Product Rule states:

f'(x)g(x)+f(x)g'(x)

So we can now set it up like this:

x^2y+3xy^2-x=5

2xy+x^2dy/dx+3y^2+6xydy/dx-1=0

Now that we have the derivative, we need to clean up the answer a bit. We need to move each term without the dy/dx to the other side. So we get:

x^2dy/dx+6xydy/dx=-2xy-3y^2+1

Now we factor out a dy/dx:

dy/dx(x^2+6xy)=-2xy-3y^2+1

Divide:

dy/dx=(-2xy-3y^2+1)/(x^2+6xy)

There is your answer

Mar 9, 2018

dy/dx = (1 - 2xy - 3y^2)/(x^2 + 6xy)

Explanation:

We have:

x^2y+3xy^2-x=5

Differentiating wrt x by applying the product rule we get:

x^2(d/dx y) + (d/dx x^2)y + (3x)(d/dx y^2) + (d/dx 3x)y^2 -d/dx x = d/dx 5

:. x^2(d/dx y) + (2x)y + (3x)(d/dx y^2) + (3)y^2 -1 = 0

Then we apply the chain rule to perform the implicit differentiation:

x^2(dy/dx) + (2x)y + (3x)(2y dy/dx) + (3)y^2 -1 = 0

Now, we collect terms and solve for dy/dx:

(x^2 + 6xy)(dy/dx) = 1 - 2xy - 3y^2

:. dy/dx = (1 - 2xy - 3y^2)/(x^2 + 6xy)