Find dy/dx e^(x^2+y)=x+y

Problem

e(x2+y)=x+y

Solution

1. Differentiate both sides with respect to x using the chain rule on the left side and the sum rule on the right side.

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

2. Apply the chain rule to the exponential function, noting that y is a function of x

e(x2+y)⋅d(x2+y)/d(x)=1+d(y)/d(x)

3. Differentiate the exponent term by term.

e(x2+y)⋅(2*x+d(y)/d(x))=1+d(y)/d(x)

4. Distribute the exponential term to isolate the terms containing d(y)/d(x)

2*x*e(x2+y)+e(x2+y)d(y)/d(x)=1+d(y)/d(x)

5. Group all terms involving d(y)/d(x) on one side of the equation and the remaining terms on the other.

e(x2+y)d(y)/d(x)−d(y)/d(x)=1−2*x*e(x2+y)

6. Factor out d(y)/d(x) from the left side.

d(y)/d(x)*(e(x2+y)−1)=1−2*x*e(x2+y)

7. Solve for dy/dx by dividing both sides by the factor (e(x2+y)−1)

d(y)/d(x)=(1−2*x*e(x2+y))/(e(x2+y)−1)

Final Answer

d(y)/d(x)=(1−2*x*e(x2+y))/(e(x2+y)−1)

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