Find dy/dx cos(x^2)=xe^y

Problem

cos(x2)=x*ey

Solution

1. Differentiate both sides with respect to x using the chain rule and the product rule.

d(cos(x2))/d(x)=(d(x)*ey)/d(x)

2. Apply the chain rule to the left side, where the derivative of cos(u) is −sin(u)⋅d(u)/d(x)

−2*x*sin(x2)=(d(x)*ey)/d(x)

3. Apply the product rule to the right side, treating y as a function of x such that d(ey)/d(x)=eyd(y)/d(x)

−2*x*sin(x2)=ey+x*eyd(y)/d(x)

4. Isolate the term containing d(y)/d(x) by subtracting ey from both sides.

−2*x*sin(x2)−ey=x*eyd(y)/d(x)

5. Solve for dy/dx by dividing both sides by x*ey

d(y)/d(x)=(−2*x*sin(x2)−ey)/(x*ey)

6. Simplify the expression by splitting the fraction.

d(y)/d(x)=−(2*sin(x2))/(ey)−1/x

Final Answer

d(y)/d(x)=(−2*x*sin(x2)−ey)/(x*ey)

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