What is the derivative of (e^x/((1+e^{2x))^{3/2}}) ?

The derivative of (e^x/((1+e^{2x))^{3/2}}) is (e^x(-2e^{2x}+1))/((1+e^{2x))^{5/2}}

What is the first derivative of (e^x/((1+e^{2x))^{3/2}}) ?

The first derivative of (e^x/((1+e^{2x))^{3/2}}) is (e^x(-2e^{2x}+1))/((1+e^{2x))^{5/2}}

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\frac{d}{d x} (\frac{e ^{x}}{( 1 + e ^{2 x} ) ^{\frac{3}{2}}})

\frac{e ^{x} ( - 2 e ^{2 x} + 1 )}{( 1 + e ^{2 x} ) ^{\frac{5}{2}}}

Solution steps

\frac{d}{d x} (\frac{e ^{x}}{( 1 + e ^{2 x} ) ^{\frac{3}{2}}})

Apply the Quotient Rule:

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

= \frac{\frac{d}{d x} ( e ^{x} ) ( 1 + e ^{2 x} ) ^{\frac{3}{2}} - \frac{d}{d x} ( ( 1 + e ^{2 x} ) ^{\frac{3}{2}} ) e ^{x}}{( ( 1 + e ^{2 x} ) ^{\frac{3}{2}} ) ^{2}}

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

\frac{d}{d x} ((1 + e^{2 x})^{\frac{3}{2}}) = 3 e^{2 x} (1 + e^{2 x})^{\frac{1}{2}}

= \frac{e ^{x} ( 1 + e ^{2 x} ) ^{\frac{3}{2}} - 3 e ^{2 x} ( 1 + e ^{2 x} ) ^{\frac{1}{2}} e ^{x}}{( ( 1 + e ^{2 x} ) ^{\frac{3}{2}} ) ^{2}}

Simplify

\frac{e ^{x} ( 1 + e ^{2 x} ) ^{\frac{3}{2}} - 3 e ^{2 x} ( 1 + e ^{2 x} ) ^{\frac{1}{2}} e ^{x}}{( ( 1 + e ^{2 x} ) ^{\frac{3}{2}} ) ^{2}} : \frac{e ^{x} ( - 2 e ^{2 x} + 1 )}{( 1 + e ^{2 x} ) ^{\frac{5}{2}}}

= \frac{e ^{x} ( - 2 e ^{2 x} + 1 )}{( 1 + e ^{2 x} ) ^{\frac{5}{2}}}

\int \frac{( x ^{2} - a ^{2} ) ^{\frac{3}{2}}}{x} d x

\frac{d y}{d t} = \frac{y}{t + 1} + 4 t^{2} + 4 t, y (1) = 12

y = 5 x^{8} - 3 x^{5} + 2, (0, 4)

\frac{d}{d x} (θ (x)), tan (θ (x)) = \frac{x}{35}

(x) \frac{d y}{d x} + y = \frac{1}{y ^{2}}

What is the derivative of (e^x/((1+e^{2x))^{3/2}}) ?
The derivative of (e^x/((1+e^{2x))^{3/2}}) is (e^x(-2e^{2x}+1))/((1+e^{2x))^{5/2}}
What is the first derivative of (e^x/((1+e^{2x))^{3/2}}) ?
The first derivative of (e^x/((1+e^{2x))^{3/2}}) is (e^x(-2e^{2x}+1))/((1+e^{2x))^{5/2}}