What is the derivative of 1/(ax^2+bx+c) ?

The derivative of 1/(ax^2+bx+c) is -(2ax+b)/((ax^2+bx+c)^2)

What is the first derivative of 1/(ax^2+bx+c) ?

The first derivative of 1/(ax^2+bx+c) is -(2ax+b)/((ax^2+bx+c)^2)

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

- \frac{2 a x + b}{( a x ^{2} + b x + c ) ^{2}}

Solution steps

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

Apply exponent rule:

\frac{1}{a} = a^{- 1}

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

Apply the chain rule:

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

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

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

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

Simplify

- \frac{1}{( a x ^{2} + b x + c ) ^{2}} (2 a x + b) : - \frac{2 a x + b}{( a x ^{2} + b x + c ) ^{2}}

= - \frac{2 a x + b}{( a x ^{2} + b x + c ) ^{2}}

\int \frac{3}{( x ^{2} - 1 ) ^{2}} d x

\frac{\partial}{\partial x} (x + 6 y - 37)

y^{^{''}} = - a^{2} y

x \to 3 - lim (c x^{2} + 6 x)

\int (4 x^{3} + x^{2}) d x

What is the derivative of 1/(ax^2+bx+c) ?
The derivative of 1/(ax^2+bx+c) is -(2ax+b)/((ax^2+bx+c)^2)
What is the first derivative of 1/(ax^2+bx+c) ?
The first derivative of 1/(ax^2+bx+c) is -(2ax+b)/((ax^2+bx+c)^2)