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

Problem

d(cos(e^cos(t)))/d(t)

Solution

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

2. Apply the power rule to differentiate the square, which gives 2*cos(e^cos(t)) multiplied by the derivative of the inner cosine function.

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

4. Differentiate the exponential function e^v which remains e^cos(t) multiplied by the derivative of the exponent cos(t)

5. Apply the power rule to cos(t) resulting in 2*cos(t) multiplied by the derivative of cos(t)

6. Differentiate the innermost function cos(t) to get −sin(t)

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

8. Simplify the expression using the double angle identity sin(2*θ)=2*sin(θ)*cos(θ) for both the t terms and the e^cos(t) terms.

Final Answer

d(cos(e^cos(t)))/d(t)=2*sin(2*e^cos(t))*e^cos(t)*sin(t)*cos(t)

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