Find the Derivative - d/dx xe^(xy)

Problem

d()/d(x)*x*e(x*y)

Solution

1. Identify the rule needed for the expression x*e(x*y) which is the product rule since x and e(x*y) are multiplied.

2. Apply the product rule formula d()/d(x)*u*v=ud(v)/d(x)+vd(u)/d(x) where u=x and v=e(x*y)

3. Differentiate the first part u=x with respect to x

d(x)/d(x)=1

4. Apply the chain rule to differentiate v=e(x*y) with respect to x treating y as a function of x (implicit differentiation).

d(e(x*y))/d(x)=e(x*y)(d(x)*y)/d(x)

5. Apply the product rule again to the exponent x*y

(d(x)*y)/d(x)=xd(y)/d(x)+yd(x)/d(x)

(d(x)*y)/d(x)=xd(y)/d(x)+y

6. Combine the results into the derivative of e(x*y)

d(e(x*y))/d(x)=e(x*y)*(xd(y)/d(x)+y)

7. Substitute all components back into the main product rule expression.

d()/d(x)*x*e(x*y)=x*(e(x*y)*(xd(y)/d(x)+y))+e(1)(x*y)

8. Simplify the expression by distributing x and factoring out e(x*y)

d()/d(x)*x*e(x*y)=x2*e(x*y)d(y)/d(x)+x*y*e(x*y)+e(x*y)

d()/d(x)*x*e(x*y)=e(x*y)*(x2d(y)/d(x)+x*y+1)

Final Answer

d()/d(x)*x*e(x*y)=e(x*y)*(x2d(y)/d(x)+x*y+1)

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