If y = x(e)^xy ,then dy/dx =.............

To find the derivative of the function y = x(e)^xy, we will use both the product rule and the chain rule.

The product rule states that the derivative of two functions multiplied together is the first function times the derivative of the second function plus the second function times the derivative of the first function.

The chain rule states that the derivative of a composite function is the derivative of the outer function times the derivative of the inner function.

Step 1: Identify the two functions in the product. Here, they are x and (e)^xy.

Step 2: Find the derivative of x, which is 1.

Step 3: Find the derivative of (e)^xy. This is a composite function, so we use the chain rule. The outer function is e^u and its derivative is e^u. The inner function is xy, and its derivative (using the product rule) is x(dy/dx) + y. So, the derivative of (e)^xy is e^xy * (x(dy/dx) + y).

Step 4: Apply the product rule. The derivative of y = x(e)^xy is x * derivative of (e)^xy + (e)^xy * derivative of x. Substituting from steps 2 and 3, we get x * e^xy * (x(dy/dx) + y) + (e)^xy * 1.

Step 5: Simplify the expression. We get x^2 * e^xy * (dy/dx) + x * y * e^xy + e^xy.

Step 6: Solve for dy/dx. We get dy/dx = (e^xy - x * y * e^xy) / (x^2 * e^xy).

So, dy/dx = (e^xy - x * y * e^xy) / (x^2 * e^xy).