Find dy/dt y=e^(tan(pit))

Problem

d()/d(t)*etan(π*t)

Solution

1. Identify the outer function as eu and the inner function as u=tan(π*t)

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

3. Differentiate the inner function u=tan(π*t) using the chain rule again.

4. Apply the derivative of the tangent function, where d(tan(v))/d(t)=sec2(v)⋅d(v)/d(t)

5. Differentiate the innermost argument v=π*t with respect to t which results in π

6. Combine all parts of the chain rule to find the final derivative.

Final Answer

d(y)/d(t)=π*sec2(π*t)*etan(π*t)

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