Find dy/dx by implicit differentiation.
X^2 + y^2 = tan(xy)
To find dy/dx by implicit differentiation, follow these steps:
Step 1: Write down the equation.
The given equation is:
x^2 + y^2 = tan(xy)
Step 2: Differentiate both sides of the equation with respect to x.
Differentiate the left-hand side with respect to x using the chain rule and the right-hand side using the product rule.
For the left-hand side, we have:
d/dx (x^2 + y^2) = d/dx (tan(xy))
Using the chain rule on the left side:
2x + 2y * dy/dx = d/dx (tan(xy))
Using the product rule on the right side:
2x + 2y * dy/dx = sec^2(xy) * (y + x * dy/dx)
Step 3: Group the terms involving dy/dx on one side of the equation.
To isolate dy/dx, move the terms involving dy/dx to one side of the equation while keeping other terms on the other side.
2x + 2y * dy/dx - sec^2(xy) * x * dy/dx = sec^2(xy) * y
Step 4: Factor out dy/dx and simplify.
Factor out dy/dx on the left-hand side and simplify the right-hand side.
dy/dx * (2y - sec^2(xy) * x) = sec^2(xy) * y - 2x
Step 5: Divide both sides by (2y - sec^2(xy) * x).
Divide both sides of the equation by (2y - sec^2(xy) * x) to solve for dy/dx.
dy/dx = (sec^2(xy) * y - 2x) / (2y - sec^2(xy) * x)
So, the expression for dy/dx in terms of x and y is:
dy/dx = (sec^2(xy) * y - 2x) / (2y - sec^2(xy) * x)