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

Problem

d(cos(ecos(t)))/d(t)

Solution

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

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

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

4. Differentiate the exponential function ev which remains ecos(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 ecos(t) terms.

Final Answer

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

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