What is the derivative of-(x^3+8x^2+19x+12/((x+20))) ?

The derivative of-(x^3+8x^2+19x+12/((x+20))) is -(2x^3+68x^2+320x+368)/((x+20)^2)

What is the first derivative of-(x^3+8x^2+19x+12/((x+20))) ?

The first derivative of-(x^3+8x^2+19x+12/((x+20))) is -(2x^3+68x^2+320x+368)/((x+20)^2)

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\frac{d}{d x} (- \frac{x ^{3} + 8 x ^{2} + 19 x + 12}{( x + 20 )})

- \frac{2 x ^{3} + 68 x ^{2} + 320 x + 368}{( x + 20 ) ^{2}}

Solution steps

\frac{d}{d x} (- \frac{x ^{3} + 8 x ^{2} + 19 x + 12}{( x + 20 )})

Take the constant out:

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

= - \frac{d}{d x} (\frac{x ^{3} + 8 x ^{2} + 19 x + 12}{x + 20})

Apply the Quotient Rule:

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

= - \frac{\frac{d}{d x} ( x ^{3} + 8 x ^{2} + 19 x + 12 ) ( x + 20 ) - \frac{d}{d x} ( x + 20 ) ( x ^{3} + 8 x ^{2} + 19 x + 12 )}{( x + 20 ) ^{2}}

\frac{d}{d x} (x^{3} + 8 x^{2} + 19 x + 12) = 3 x^{2} + 16 x + 19

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

= - \frac{( 3 x ^{2} + 16 x + 19 ) ( x + 20 ) - 1 \cdot ( x ^{3} + 8 x ^{2} + 19 x + 12 )}{( x + 20 ) ^{2}}

Simplify

- \frac{( 3 x ^{2} + 16 x + 19 ) ( x + 20 ) - 1 \cdot ( x ^{3} + 8 x ^{2} + 19 x + 12 )}{( x + 20 ) ^{2}} : - \frac{2 x ^{3} + 68 x ^{2} + 320 x + 368}{( x + 20 ) ^{2}}

= - \frac{2 x ^{3} + 68 x ^{2} + 320 x + 368}{( x + 20 ) ^{2}}

What is the derivative of-(x^3+8x^2+19x+12/((x+20))) ?
The derivative of-(x^3+8x^2+19x+12/((x+20))) is -(2x^3+68x^2+320x+368)/((x+20)^2)
What is the first derivative of-(x^3+8x^2+19x+12/((x+20))) ?
The first derivative of-(x^3+8x^2+19x+12/((x+20))) is -(2x^3+68x^2+320x+368)/((x+20)^2)