Find the Derivative - d/dt e^(7tsin(2t))

Problem

d(e(7*t*sin(2*t)))/d(t)

Solution

1. Identify the outer function as the exponential function eu and the inner function as u=7*t*sin(2*t)

2. Apply the chain rule for exponential functions, which states d(eu)/d(t)=eud(u)/d(t)

3. Apply the product rule to differentiate u=7*t*sin(2*t) where the two parts are 7*t and sin(2*t)

4. Apply the chain rule to differentiate sin(2*t) which results in 2*cos(2*t)

5. Calculate the derivative of the inner function:

(d(7)*t*sin(2*t))/d(t)=7*sin(2*t)+7*t*(2*cos(2*t))

6. Simplify the expression for the inner derivative:

(d(7)*t*sin(2*t))/d(t)=7*sin(2*t)+14*t*cos(2*t)

7. Combine the results back into the chain rule formula.

Final Answer

d(e(7*t*sin(2*t)))/d(t)=(7*sin(2*t)+14*t*cos(2*t))*e(7*t*sin(2*t))

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