What is the derivative of (6e^x/((6+e^x)^2)) ?

The derivative of (6e^x/((6+e^x)^2)) is (6e^x(-e^x+6))/((6+e^x)^3)

What is the first derivative of (6e^x/((6+e^x)^2)) ?

The first derivative of (6e^x/((6+e^x)^2)) is (6e^x(-e^x+6))/((6+e^x)^3)

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\frac{d}{d x} (\frac{6 e ^{x}}{( 6 + e ^{x} ) ^{2}})

\frac{6 e ^{x} ( - e ^{x} + 6 )}{( 6 + e ^{x} ) ^{3}}

Solution steps

\frac{d}{d x} (\frac{6 e ^{x}}{( 6 + e ^{x} ) ^{2}})

Take the constant out:

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

= 6 \frac{d}{d x} (\frac{e ^{x}}{( 6 + e ^{x} ) ^{2}})

Apply the Quotient Rule:

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

= 6 \cdot \frac{\frac{d}{d x} ( e ^{x} ) ( 6 + e ^{x} ) ^{2} - \frac{d}{d x} ( ( 6 + e ^{x} ) ^{2} ) e ^{x}}{( ( 6 + e ^{x} ) ^{2} ) ^{2}}

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

\frac{d}{d x} ((6 + e^{x})^{2}) = 2 (6 + e^{x}) e^{x}

= 6 \cdot \frac{e ^{x} ( 6 + e ^{x} ) ^{2} - 2 ( 6 + e ^{x} ) e ^{x} e ^{x}}{( ( 6 + e ^{x} ) ^{2} ) ^{2}}

Simplify

6 \cdot \frac{e ^{x} ( 6 + e ^{x} ) ^{2} - 2 ( 6 + e ^{x} ) e ^{x} e ^{x}}{( ( 6 + e ^{x} ) ^{2} ) ^{2}} : \frac{6 e ^{x} ( - e ^{x} + 6 )}{( 6 + e ^{x} ) ^{3}}

= \frac{6 e ^{x} ( - e ^{x} + 6 )}{( 6 + e ^{x} ) ^{3}}

2 arctan (x)

x \to - 3 lim (7 x^{2})

\int x e^{5 x^{2}} d x

x^{2} + x y + 2 y^{2} = 4

What is the derivative of (6e^x/((6+e^x)^2)) ?
The derivative of (6e^x/((6+e^x)^2)) is (6e^x(-e^x+6))/((6+e^x)^3)
What is the first derivative of (6e^x/((6+e^x)^2)) ?
The first derivative of (6e^x/((6+e^x)^2)) is (6e^x(-e^x+6))/((6+e^x)^3)