What is the d/(dt)((e^tθ)/((1-e^t(1-θ))^2)) ?

The d/(dt)((e^tθ)/((1-e^t(1-θ))^2)) is (e^tθ(e^t(-θ+1)+1))/((1-e^t(-θ+1))^3)

What is the first d/(dt)((e^tθ)/((1-e^t(1-θ))^2)) ?

The first d/(dt)((e^tθ)/((1-e^t(1-θ))^2)) is (e^tθ(e^t(-θ+1)+1))/((1-e^t(-θ+1))^3)

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\frac{d}{d t} (\frac{e ^{t} θ}{( 1 - e ^{t} ( 1 - θ ) ) ^{2}})

\frac{e ^{t} θ ( e ^{t} ( - θ + 1 ) + 1 )}{( 1 - e ^{t} ( - θ + 1 ) ) ^{3}}

Solution steps

\frac{d}{d t} (\frac{e ^{t} θ}{( 1 - e ^{t} ( 1 - θ ) ) ^{2}})

Treat

θ

as a constant

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= θ \frac{d}{d t} (\frac{e ^{t}}{( 1 - e ^{t} ( 1 - θ ) ) ^{2}})

Apply the Quotient Rule:

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

= θ \frac{\frac{d}{d t} ( e ^{t} ) ( 1 - e ^{t} ( 1 - θ ) ) ^{2} - \frac{d}{d t} ( ( 1 - e ^{t} ( 1 - θ ) ) ^{2} ) e ^{t}}{( ( 1 - e ^{t} ( 1 - θ ) ) ^{2} ) ^{2}}

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

\frac{d}{d t} ((1 - e^{t} (1 - θ))^{2}) = - 2 e^{t} (1 - e^{t} (1 - θ)) (- θ + 1)

= θ \frac{e ^{t} ( 1 - e ^{t} ( 1 - θ ) ) ^{2} - ( - 2 e ^{t} ( 1 - e ^{t} ( 1 - θ ) ) ( - θ + 1 ) ) e ^{t}}{( ( 1 - e ^{t} ( 1 - θ ) ) ^{2} ) ^{2}}

Simplify

θ \frac{e ^{t} ( 1 - e ^{t} ( 1 - θ ) ) ^{2} - ( - 2 e ^{t} ( 1 - e ^{t} ( 1 - θ ) ) ( - θ + 1 ) ) e ^{t}}{( ( 1 - e ^{t} ( 1 - θ ) ) ^{2} ) ^{2}} : \frac{e ^{t} θ ( e ^{t} ( - θ + 1 ) + 1 )}{( 1 - e ^{t} ( - θ + 1 ) ) ^{3}}

= \frac{e ^{t} θ ( e ^{t} ( - θ + 1 ) + 1 )}{( 1 - e ^{t} ( - θ + 1 ) ) ^{3}}

\frac{\partial}{\partial t} (e^{- t} sin (t))

\int (a^{5 x} + e^{5 x}) d x

\int \frac{1}{4 - y ^{2}} d y

\int sin^{8} (3 x) cos^{5} (3 x) d x

\frac{d}{d x} (10 x^{4})

What is the d/(dt)((e^tθ)/((1-e^t(1-θ))^2)) ?
The d/(dt)((e^tθ)/((1-e^t(1-θ))^2)) is (e^tθ(e^t(-θ+1)+1))/((1-e^t(-θ+1))^3)
What is the first d/(dt)((e^tθ)/((1-e^t(1-θ))^2)) ?
The first d/(dt)((e^tθ)/((1-e^t(1-θ))^2)) is (e^tθ(e^t(-θ+1)+1))/((1-e^t(-θ+1))^3)