Find the Derivative - d/dt e^(3tsin(2t))

Problem

d(e(3*t*sin(2*t)))/d(t)

Solution

1. Identify the outer function as eu and the inner function as u=3*t*sin(2*t)

2. Apply the chain rule, which states that the derivative of eu(t) is eu(t)⋅d(u)/d(t)

3. Apply the product rule to find d(u)/d(t) for the expression 3*t⋅sin(2*t)

4. Differentiate the first part of the product: (d(3)*t)/d(t)=3

5. Differentiate the second part of the product using the chain rule: d(sin(2*t))/d(t)=cos(2*t)⋅2=2*cos(2*t)

6. Combine the product rule results: d(u)/d(t)=3⋅sin(2*t)+3*t⋅2*cos(2*t)

7. Simplify the inner derivative expression: d(u)/d(t)=3*sin(2*t)+6*t*cos(2*t)

8. Substitute the inner derivative back into the chain rule formula.

Final Answer

d(e(3*t*sin(2*t)))/d(t)=e(3*t*sin(2*t))*(3*sin(2*t)+6*t*cos(2*t))

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