What is the derivative of (e^{ax^2+bx+c}/x) ?

The derivative of (e^{ax^2+bx+c}/x) is (e^{ax^2+bx+c}x(2ax+b)-e^{ax^2+bx+c})/(x^2)

What is the first derivative of (e^{ax^2+bx+c}/x) ?

The first derivative of (e^{ax^2+bx+c}/x) is (e^{ax^2+bx+c}x(2ax+b)-e^{ax^2+bx+c})/(x^2)

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

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

Solution steps

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

Apply the Quotient Rule:

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

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

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

\frac{d x}{d x} = 1

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

Simplify

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

e \to 1 lim (5 ln^{2} (e))

x \to + 0 lim ((1 - \frac{1}{x}) x)

\frac{\partial}{\partial y} (x + y + 1)

\int \frac{1}{x ^{2022}} d x

\frac{d}{d t} (7 t^{3})

What is the derivative of (e^{ax^2+bx+c}/x) ?
The derivative of (e^{ax^2+bx+c}/x) is (e^{ax^2+bx+c}x(2ax+b)-e^{ax^2+bx+c})/(x^2)
What is the first derivative of (e^{ax^2+bx+c}/x) ?
The first derivative of (e^{ax^2+bx+c}/x) is (e^{ax^2+bx+c}x(2ax+b)-e^{ax^2+bx+c})/(x^2)