# How do you implicitly differentiate #y= xy^2 + x ^2 e^(x -2y) #?

Use the product rule again:

Distribute the x^2:

By signing up, you agree to our Terms of Service and Privacy Policy

By signing up, you agree to our Terms of Service and Privacy Policy

# dy/dx = (y^2+(x^2+2x)e^(x-2y))/(1+2x^2e^(x-2y)-2xy)#

Using the Implicit Function Theorem we have:

Where:

We have:

Let:

Then the partial derivatives are:

And:

And so:

By signing up, you agree to our Terms of Service and Privacy Policy

To implicitly differentiate ( y = xy^2 + x^2e^{x - 2y} ):

- Differentiate each term with respect to ( x ).
- Apply the product rule for the term ( xy^2 ) and the chain rule for the term ( x^2e^{x - 2y} ).
- Simplify the resulting expression.

The implicit differentiation of the given function yields:

[ \frac{dy}{dx} = y^2 + 2xy\frac{dy}{dx} + 2xe^{x - 2y} - 2x\frac{dy}{dx}e^{x - 2y} ]

Now, solve for ( \frac{dy}{dx} ):

[ \frac{dy}{dx} - 2xy\frac{dy}{dx} + 2x\frac{dy}{dx}e^{x - 2y} = y^2 + 2xe^{x - 2y} ]

[ \frac{dy}{dx}(1 - 2xy + 2xe^{x - 2y}) = y^2 + 2xe^{x - 2y} ]

[ \frac{dy}{dx} = \frac{y^2 + 2xe^{x - 2y}}{1 - 2xy + 2xe^{x - 2y}} ]

By signing up, you agree to our Terms of Service and Privacy Policy

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- How do you find the derivative of #sqrt(1/x^3)#?
- How do you find the derivative of #4x(x+x^(1/2))^(1/2)#?
- What is the derivative of #y=1/2(x^2-x^-2) #?
- How do you find the derivative of exponential function #f(x)= 2 / (e^x + e^-x)^3#?
- How do you differentiate #f(x)=(2+x )sqrt(2-3x# using the product rule?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7