What is the derivative of (ax^2+bx+1/(x^2-6x+8)) ?

The derivative of (ax^2+bx+1/(x^2-6x+8)) is (-6ax^2-bx^2+16ax-2x+8b+6)/((x^2-6x+8)^2)

What is the first derivative of (ax^2+bx+1/(x^2-6x+8)) ?

The first derivative of (ax^2+bx+1/(x^2-6x+8)) is (-6ax^2-bx^2+16ax-2x+8b+6)/((x^2-6x+8)^2)

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\frac{d}{d x} (\frac{a x ^{2} + b x + 1}{x ^{2} - 6 x + 8})

\frac{- 6 a x ^{2} - b x ^{2} + 16 a x - 2 x + 8 b + 6}{( x ^{2} - 6 x + 8 ) ^{2}}

Solution steps

\frac{d}{d x} (\frac{a x ^{2} + b x + 1}{x ^{2} - 6 x + 8})

Apply the Quotient Rule:

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

= \frac{\frac{d}{d x} ( a x ^{2} + b x + 1 ) ( x ^{2} - 6 x + 8 ) - \frac{d}{d x} ( x ^{2} - 6 x + 8 ) ( a x ^{2} + b x + 1 )}{( x ^{2} - 6 x + 8 ) ^{2}}

\frac{d}{d x} (a x^{2} + b x + 1) = 2 a x + b

\frac{d}{d x} (x^{2} - 6 x + 8) = 2 x - 6

= \frac{( 2 a x + b ) ( x ^{2} - 6 x + 8 ) - ( 2 x - 6 ) ( a x ^{2} + b x + 1 )}{( x ^{2} - 6 x + 8 ) ^{2}}

Expand

(2 a x + b) (x^{2} - 6 x + 8) - (2 x - 6) (a x^{2} + b x + 1) : - 6 a x^{2} - b x^{2} + 16 a x - 2 x + 8 b + 6

= \frac{- 6 a x ^{2} - b x ^{2} + 16 a x - 2 x + 8 b + 6}{( x ^{2} - 6 x + 8 ) ^{2}}

\frac{d ^{2}}{d x ^{2}} ((3 + 2 x) e^{- 5 x})

f (x) = x^{2} + 1, at x = 1

x \to 2 lim (3 x^{2} - 5 x + 1)

\int_{0}^{1} x^{5} d x

x^{3} y^{^{'''}} - 3 x^{2} y^{^{''}} + 6 x y^{^{'}} - 6 y = 0

What is the derivative of (ax^2+bx+1/(x^2-6x+8)) ?
The derivative of (ax^2+bx+1/(x^2-6x+8)) is (-6ax^2-bx^2+16ax-2x+8b+6)/((x^2-6x+8)^2)
What is the first derivative of (ax^2+bx+1/(x^2-6x+8)) ?
The first derivative of (ax^2+bx+1/(x^2-6x+8)) is (-6ax^2-bx^2+16ax-2x+8b+6)/((x^2-6x+8)^2)