What is the derivative of (e^xcos(x)/(e^x-1)) ?

The derivative of (e^xcos(x)/(e^x-1)) is (-e^xcos(x)-e^{2x}sin(x)+e^xsin(x))/((e^x-1)^2)

What is the first derivative of (e^xcos(x)/(e^x-1)) ?

The first derivative of (e^xcos(x)/(e^x-1)) is (-e^xcos(x)-e^{2x}sin(x)+e^xsin(x))/((e^x-1)^2)

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\frac{d}{d x} (\frac{e ^{x} cos ( x )}{e ^{x} - 1})

\frac{- e ^{x} cos ( x ) - e ^{2 x} sin ( x ) + e ^{x} sin ( x )}{( e ^{x} - 1 ) ^{2}}

Solution steps

\frac{d}{d x} (\frac{e ^{x} cos ( x )}{e ^{x} - 1})

Apply the Quotient Rule:

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

= \frac{\frac{d}{d x} ( e ^{x} cos ( x ) ) ( e ^{x} - 1 ) - \frac{d}{d x} ( e ^{x} - 1 ) e ^{x} cos ( x )}{( e ^{x} - 1 ) ^{2}}

\frac{d}{d x} (e^{x} cos (x)) = e^{x} cos (x) - e^{x} sin (x)

\frac{d}{d x} (e^{x} - 1) = e^{x}

= \frac{( e ^{x} cos ( x ) - e ^{x} sin ( x ) ) ( e ^{x} - 1 ) - e ^{x} e ^{x} cos ( x )}{( e ^{x} - 1 ) ^{2}}

Simplify

\frac{( e ^{x} cos ( x ) - e ^{x} sin ( x ) ) ( e ^{x} - 1 ) - e ^{x} e ^{x} cos ( x )}{( e ^{x} - 1 ) ^{2}} : \frac{- e ^{x} cos ( x ) - e ^{2 x} sin ( x ) + e ^{x} sin ( x )}{( e ^{x} - 1 ) ^{2}}

= \frac{- e ^{x} cos ( x ) - e ^{2 x} sin ( x ) + e ^{x} sin ( x )}{( e ^{x} - 1 ) ^{2}}

\frac{d}{d t} (4 cos (2 t))

\frac{d y}{d x} = (\frac{y}{y - x})

\frac{d}{d x} (x (x + 4)^{\frac{1}{3}})

x \to 0 - lim (x^{3} + 2 x + 2)

y^{^{'}} = \frac{x}{y} + \frac{y}{x}

What is the derivative of (e^xcos(x)/(e^x-1)) ?
The derivative of (e^xcos(x)/(e^x-1)) is (-e^xcos(x)-e^{2x}sin(x)+e^xsin(x))/((e^x-1)^2)
What is the first derivative of (e^xcos(x)/(e^x-1)) ?
The first derivative of (e^xcos(x)/(e^x-1)) is (-e^xcos(x)-e^{2x}sin(x)+e^xsin(x))/((e^x-1)^2)