What is the derivative of (cot(t))/(e^t) ?

The derivative of (cot(t))/(e^t) is -(csc^2(t)+cot(t))/(e^t)

What is the first derivative of (cot(t))/(e^t) ?

The first derivative of (cot(t))/(e^t) is -(csc^2(t)+cot(t))/(e^t)

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derivative of

\frac{cot ( t )}{e ^{t}}

- \frac{csc ^{2} ( t ) + cot ( t )}{e ^{t}}

Solution steps

\frac{d}{d t} (\frac{cot ( t )}{e ^{t}})

Apply the Quotient Rule:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

= \frac{\frac{d}{d t} ( cot ( t ) ) e ^{t} - \frac{d}{d t} ( e ^{t} ) cot ( t )}{( e ^{t} ) ^{2}}

\frac{d}{d t} (cot (t)) = - csc^{2} (t)

\frac{d}{d t} (e^{t}) = e^{t}

= \frac{( - csc ^{2} ( t ) ) e ^{t} - e ^{t} cot ( t )}{( e ^{t} ) ^{2}}

Simplify

\frac{( - csc ^{2} ( t ) ) e ^{t} - e ^{t} cot ( t )}{( e ^{t} ) ^{2}} : - \frac{csc ^{2} ( t ) + cot ( t )}{e ^{t}}

= - \frac{csc ^{2} ( t ) + cot ( t )}{e ^{t}}

What is the derivative of (cot(t))/(e^t) ?
The derivative of (cot(t))/(e^t) is -(csc^2(t)+cot(t))/(e^t)
What is the first derivative of (cot(t))/(e^t) ?
The first derivative of (cot(t))/(e^t) is -(csc^2(t)+cot(t))/(e^t)