Find dy/dx e^(2x)=sin(x+3y)

Problem

e(2*x)=sin(x+3*y)

Solution

1. Differentiate both sides with respect to x using the chain rule for both the exponential and trigonometric functions.

2. Apply the chain rule to the left side, where the derivative of e(2*x) is e(2*x)⋅2

d(e(2*x))/d(x)=2*e(2*x)

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

d(sin(x+3*y))/d(x)=cos(x+3*y)⋅(1+3d(y)/d(x))

4. Equate the derivatives obtained from both sides of the original equation.

2*e(2*x)=cos(x+3*y)⋅(1+3d(y)/d(x))

5. Distribute the cosine term to isolate the term containing d(y)/d(x)

2*e(2*x)=cos(x+3*y)+3*cos(x+3*y)d(y)/d(x)

6. Isolate the derivative by subtracting cos(x+3*y) from both sides and then dividing by 3*cos(x+3*y)

2*e(2*x)−cos(x+3*y)=3*cos(x+3*y)d(y)/d(x)

7. Solve for dy/dx to find the final expression.

d(y)/d(x)=(2*e(2*x)−cos(x+3*y))/(3*cos(x+3*y))

Final Answer

d(y)/d(x)=(2*e(2*x)−cos(x+3*y))/(3*cos(x+3*y))

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