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

Problem

d()/d(t)*sin(e^sin(t))

Solution

1. Identify the outer function as a power function of the form u^2 where u=sin(e^sin(t))

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

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

4. Differentiate the exponential function e^sin(t) which remains e^sin(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(e^sin(t))=2*sin(e^sin(t))*cos(e^sin(t))*e^sin(t)*sin(2*t)

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