What is the derivative of ln((4x/(x^2+5))) ?

The derivative of ln((4x/(x^2+5))) is (-x^2+5)/(x(x^2+5))

What is the first derivative of ln((4x/(x^2+5))) ?

The first derivative of ln((4x/(x^2+5))) is (-x^2+5)/(x(x^2+5))

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\frac{d}{d x} (ln (\frac{4 x}{x ^{2} + 5}))

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

Solution steps

\frac{d}{d x} (ln (\frac{4 x}{x ^{2} + 5}))

Apply the chain rule:

\frac{1}{\frac{4 x}{x ^{2} + 5}} \frac{d}{d x} (\frac{4 x}{x ^{2} + 5})

= \frac{1}{\frac{4 x}{x ^{2} + 5}} \frac{d}{d x} (\frac{4 x}{x ^{2} + 5})

\frac{d}{d x} (\frac{4 x}{x ^{2} + 5}) = \frac{4 ( - x ^{2} + 5 )}{( x ^{2} + 5 ) ^{2}}

= \frac{1}{\frac{4 x}{x ^{2} + 5}} \cdot \frac{4 ( - x ^{2} + 5 )}{( x ^{2} + 5 ) ^{2}}

Simplify

\frac{1}{\frac{4 x}{x ^{2} + 5}} \cdot \frac{4 ( - x ^{2} + 5 )}{( x ^{2} + 5 ) ^{2}} : \frac{- x ^{2} + 5}{x ( x ^{2} + 5 )}

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

\frac{d}{d x} (sin (x + y))

x \to 2 lim ((x^{2} - 4)^{- 1} sin (x - 2))

y^{^{''}} + 4 y = 6 sin (3 t)

\int 6 sin^{4} (x) cos (x) d x

x^{3} d x + y^{3} d y = 0

What is the derivative of ln((4x/(x^2+5))) ?
The derivative of ln((4x/(x^2+5))) is (-x^2+5)/(x(x^2+5))
What is the first derivative of ln((4x/(x^2+5))) ?
The first derivative of ln((4x/(x^2+5))) is (-x^2+5)/(x(x^2+5))