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

Problem

cos(y2)+x=ey

Solution

1. Differentiate both sides with respect to x treating y as a function of x and applying the chain rule.

d(cos(y2))/d(x)+d(x)/d(x)=d(ey)/d(x)

2. Apply the chain rule to the trigonometric and exponential terms.

−sin(y2)⋅2*yd(y)/d(x)+1=eyd(y)/d(x)

3. Group the terms containing d(y)/d(x) on one side of the equation.

1=eyd(y)/d(x)+2*y*sin(y2)d(y)/d(x)

4. Factor out the common term d(y)/d(x)

1=d(y)/d(x)*(ey+2*y*sin(y2))

5. Solve for the derivative by dividing both sides by the expression in parentheses.

d(y)/d(x)=1/(ey+2*y*sin(y2))

Final Answer

d(y)/d(x)=1/(ey+2*y*sin(y2))

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