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