Question

find \\( \frac { d s } { d t } \\).
\\( s = 3 \cot t - e ^ { - t } \\)
\\( \frac { d s } { d t } = \\)

Explanation:

Step1: Differentiate \(3\cot t\)

The derivative of \(\cot t\) is \(-\csc^{2}t\). Using the constant multiple rule, the derivative of \(3\cot t\) is \(3\times(-\csc^{2}t)=- 3\csc^{2}t\).

Step2: Differentiate \(-e^{-t}\)

Using the chain rule, if \(u = -t\), then \(\frac{d}{dt}(-e^{-t})=-e^{-t}\times(-1)=e^{-t}\).

Step3: Combine the derivatives

By the sum - difference rule of differentiation \(\frac{ds}{dt}=\frac{d}{dt}(3\cot t)-\frac{d}{dt}(e^{-t})\).

Answer:

\(\frac{ds}{dt}=-3\csc^{2}t + e^{-t}\)