How do you use imlicit differentiation to differentiate e^(xy)?

I have the problem "use implicit differentiation to find dy/dx.
e^(xy)+ x^2 - y^2 = 10. I've gotten to the point whereI have d/dx(e^(xy)) + 2x - 2ydy/dx = 0, but I can't go any further because I don't know how to use implicit differentiation to find e/dx(e^(xy)). It's been a while since I've done this and i don't remember, can someone refresh my memory? Thanks1

e^(xy) + x^2 - y^2 = 10

e^(xy) (xdy/dx + y) + 2x - 2y dy/dx = 0

xe^(xy) dy/dx + ye^(xy) + 2x - 2y dy/dx = 0

dy/dx (xe^(xy) - 2y) = -2x - ye^(xy)

dy/dx = (2x + ye^(xy) ) / (2y - xe^(xy) )

To differentiate the function e^(xy) using implicit differentiation, you need to consider e^(xy) as a composite function and apply the chain rule.

Here's how you can proceed step by step:

Step 1: Start with the given equation e^(xy) + x^2 - y^2 = 10.

Step 2: Differentiate both sides of the equation with respect to x. We will treat y as a function of x, so we use the chain rule whenever we differentiate y terms.

On the left side, we have the composite function e^(xy), so we need to apply the chain rule. The chain rule states that if you have a function g(f(x)), its derivative with respect to x is given by g'(f(x)) * f'(x).

Differentiating e^(xy) using the chain rule:
d/dx(e^(xy)) = d/dx(xy) * d/d(xy)(e^xy), where the first term is the derivative of xy and the second term is the derivative of e^(xy) with respect to xy.

The derivative of xy with respect to x is simply y (since x is treated as a constant when differentiating with respect to x).

Step 3: Differentiate the second term, e^(xy), with respect to xy. To do this, treat xy as a single variable and differentiate e^u, where u = xy, using the chain rule.

Differentiating e^u using the chain rule:
d/dxy(e^xy) = e^xy * d/dxy(xy), where d/dxy(xy) is the derivative of xy with respect to xy. This derivative is simply 1.

So, d/dxy(e^xy) becomes e^xy * 1 = e^(xy).

Step 4: Substitute the derivatives we found back into the equation we differentiated earlier.

Returning to the differentiated equation from step 2:
d/dx(e^(xy)) + 2x - 2y * dy/dx = 0.

Substituting the derivatives we found:
y + 2x - 2y * dy/dx = 0.

Simplifying further:
-2y * dy/dx = -y - 2x.

Step 5: Solve for dy/dx by isolating it on one side of the equation.
Divide both sides of the equation by -2y to obtain:
dy/dx = (-y - 2x) / (-2y).

And there you have it! The expression for dy/dx using implicit differentiation for the given equation e^(xy) + x^2 - y^2 = 10.