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

Problem

d()/d(t)*sin(esin(t))

Solution

1. Identify the outer function as a power function of the form u2 where u=sin(esin(t))

2. Apply the power rule to differentiate the outermost layer, which gives 2*sin(esin(t)) multiplied by the derivative of the inner sine function.

3. Apply the chain rule to the sine function, resulting in cos(esin(t)) multiplied by the derivative of its argument esin(t)

4. Differentiate the exponential function esin(t) which remains esin(t) multiplied by the derivative of the exponent sin(t)

5. Apply the power rule and chain rule to the exponent sin(t) resulting in 2*sin(t)*cos(t)

6. Combine all the factors obtained from the chain rule steps.

7. Simplify the expression using the double angle identity 2*sin(t)*cos(t)=sin(2*t)

Final Answer

d()/d(t)*sin(esin(t))=2*sin(esin(t))*cos(esin(t))*esin(t)*sin(2*t)

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