Find dy/dx natural log of 9y=e^ysin(5x)

Problem

ln(9*y)=ey*sin(5*x)

Solution

1. Differentiate both sides with respect to x using implicit differentiation.

2. Apply the chain rule to the left side, noting that d(ln(u))/d(x)=1/ud(u)/d(x)

1/(9*y)⋅(d(9)*y)/d(x)=(d(ey)*sin(5*x))/d(x)

3. Simplify the left side and apply the product rule to the right side, where (d(u)*v)/d(x)=ud(v)/d(x)+vd(u)/d(x)

1/yd(y)/d(x)=eyd(sin(5*x))/d(x)+sin(5*x)d(ey)/d(x)

4. Compute the derivatives on the right side using the chain rule.

1/yd(y)/d(x)=5*ey*cos(5*x)+ey*sin(5*x)d(y)/d(x)

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

1/yd(y)/d(x)−ey*sin(5*x)d(y)/d(x)=5*ey*cos(5*x)

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

d(y)/d(x)*(1/y−ey*sin(5*x))=5*ey*cos(5*x)

7. Solve for d(y)/d(x) by dividing both sides by the expression in the parentheses.

d(y)/d(x)=(5*ey*cos(5*x))/(1/y−ey*sin(5*x))

8. Simplify the fraction by multiplying the numerator and denominator by y

d(y)/d(x)=(5*y*ey*cos(5*x))/(1−y*ey*sin(5*x))

Final Answer

d(y)/d(x)=(5*y*ey*cos(5*x))/(1−y*ey*sin(5*x))

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