What is the derivative of (-200ln(x/(40))/(x+20)) ?

The derivative of (-200ln(x/(40))/(x+20)) is -(200(x+20-xln(x/(40))))/(x(x+20)^2)

What is the first derivative of (-200ln(x/(40))/(x+20)) ?

The first derivative of (-200ln(x/(40))/(x+20)) is -(200(x+20-xln(x/(40))))/(x(x+20)^2)

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\frac{d}{d x} (\frac{- 200 ln ( \frac{x}{40} )}{x + 20})

- \frac{200 ( x + 20 - x ln ( \frac{x}{40} ) )}{x ( x + 20 ) ^{2}}

Solution steps

\frac{d}{d x} (\frac{- 200 ln ( \frac{x}{40} )}{x + 20})

Simplify

\frac{- 200 ln ( \frac{x}{40} )}{x + 20} : - \frac{200 ln ( \frac{x}{40} )}{x + 20}

= \frac{d}{d x} (- \frac{200 ln ( \frac{x}{40} )}{x + 20})

Take the constant out:

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

= - 200 \frac{d}{d x} (\frac{ln ( \frac{x}{40} )}{x + 20})

Apply the Quotient Rule:

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

= - 200 \cdot \frac{\frac{d}{d x} ( ln ( \frac{x}{40} ) ) ( x + 20 ) - \frac{d}{d x} ( x + 20 ) ln ( \frac{x}{40} )}{( x + 20 ) ^{2}}

\frac{d}{d x} (ln (\frac{x}{40})) = \frac{1}{x}

\frac{d}{d x} (x + 20) = 1

= - 200 \cdot \frac{\frac{1}{x} ( x + 20 ) - 1 \cdot ln ( \frac{x}{40} )}{( x + 20 ) ^{2}}

Simplify

- 200 \cdot \frac{\frac{1}{x} ( x + 20 ) - 1 \cdot ln ( \frac{x}{40} )}{( x + 20 ) ^{2}} : - \frac{200 ( x + 20 - x ln ( \frac{x}{40} ) )}{x ( x + 20 ) ^{2}}

= - \frac{200 ( x + 20 - x ln ( \frac{x}{40} ) )}{x ( x + 20 ) ^{2}}

\int_{0}^{15} 2 π x (15 x - x^{2}) d x

\frac{\partial}{\partial x} (ln (\frac{2 x y}{5 z}))

\int - \frac{9}{x ^{10}} d x

\frac{\partial}{\partial x} (u)

\frac{d}{d x} (4 π x)

What is the derivative of (-200ln(x/(40))/(x+20)) ?
The derivative of (-200ln(x/(40))/(x+20)) is -(200(x+20-xln(x/(40))))/(x(x+20)^2)
What is the first derivative of (-200ln(x/(40))/(x+20)) ?
The first derivative of (-200ln(x/(40))/(x+20)) is -(200(x+20-xln(x/(40))))/(x(x+20)^2)