What is the derivative of (5^x/(1+5^{2x)}) ?

The derivative of (5^x/(1+5^{2x)}) is (ln(5)*5^x-ln(5)*125^x)/((1+25^x)^2)

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

The first derivative of (5^x/(1+5^{2x)}) is (ln(5)*5^x-ln(5)*125^x)/((1+25^x)^2)

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

\frac{ln ( 5 ) \cdot 5 ^{x} - ln ( 5 ) \cdot 12 5 ^{x}}{( 1 + 2 5 ^{x} ) ^{2}}

Solution steps

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

Apply the Quotient Rule:

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

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

\frac{d}{d x} (5^{x}) = 5^{x} ln (5)

\frac{d}{d x} (1 + 5^{2 x}) = 2 ln (5) \cdot 2 5^{x}

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

Simplify

\frac{5 ^{x} ln ( 5 ) ( 1 + 5 ^{2 x} ) - 2 ln ( 5 ) \cdot 2 5 ^{x} \cdot 5 ^{x}}{( 1 + 5 ^{2 x} ) ^{2}} : \frac{ln ( 5 ) \cdot 5 ^{x} - ln ( 5 ) \cdot 12 5 ^{x}}{( 1 + 2 5 ^{x} ) ^{2}}

= \frac{ln ( 5 ) \cdot 5 ^{x} - ln ( 5 ) \cdot 12 5 ^{x}}{( 1 + 2 5 ^{x} ) ^{2}}

\int_{\frac{1}{7}}^{5} 16 x ln (7 x) d x

\frac{d}{d x} ((x)^{\frac{1}{2}})

\int_{- 2}^{2} x^{3} cos (\frac{x}{2}) + \frac{1}{2} d x

\frac{e ^{- 4 s}}{s ^{2} + 4 s + 3}

- 2 x + 3

What is the derivative of (5^x/(1+5^{2x)}) ?
The derivative of (5^x/(1+5^{2x)}) is (ln(5)*5^x-ln(5)*125^x)/((1+25^x)^2)
What is the first derivative of (5^x/(1+5^{2x)}) ?
The first derivative of (5^x/(1+5^{2x)}) is (ln(5)*5^x-ln(5)*125^x)/((1+25^x)^2)