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

Problem

cos(x^2)=x*e^y

Solution

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

d(cos(x^2))/d(x)=(d(x)*e^y)/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(x^2)=(d(x)*e^y)/d(x)

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

−2*x*sin(x^2)=e^y+x*e^yd(y)/d(x)

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

−2*x*sin(x^2)−e^y=x*e^yd(y)/d(x)

5. Solve for dy/dx by dividing both sides by x*e^y

d(y)/d(x)=(−2*x*sin(x^2)−e^y)/(x*e^y)

6. Simplify the expression by splitting the fraction.

d(y)/d(x)=−(2*sin(x^2))/(e^y)−1/x

Final Answer

d(y)/d(x)=(−2*x*sin(x^2)−e^y)/(x*e^y)

Want more problems? Check here!