What is the derivative of (e^x(x^2+2x^3)/(3x^2+x)) ?

The derivative of (e^x(x^2+2x^3)/(3x^2+x)) is (e^xx(x+2)^2(3x^3+19x^2+13x+4))/((3x+1)^2)

What is the first derivative of (e^x(x^2+2x^3)/(3x^2+x)) ?

The first derivative of (e^x(x^2+2x^3)/(3x^2+x)) is (e^xx(x+2)^2(3x^3+19x^2+13x+4))/((3x+1)^2)

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

\frac{e ^{x} x ( x + 2 ) ^{2} ( 3 x ^{3} + 19 x ^{2} + 13 x + 4 )}{( 3 x + 1 ) ^{2}}

Solution steps

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

Apply the Quotient Rule:

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

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

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

\frac{d}{d x} (3 x^{2} + x) = 6 x + 1

= \frac{( e ^{x} ( x ^{2} + 2 x ) ^{3} + 3 ( x ^{2} + 2 x ) ^{2} ( 2 x + 2 ) e ^{x} ) ( 3 x ^{2} + x ) - ( 6 x + 1 ) e ^{x} ( x ^{2} + 2 x ) ^{3}}{( 3 x ^{2} + x ) ^{2}}

Simplify

\frac{( e ^{x} ( x ^{2} + 2 x ) ^{3} + 3 ( x ^{2} + 2 x ) ^{2} ( 2 x + 2 ) e ^{x} ) ( 3 x ^{2} + x ) - ( 6 x + 1 ) e ^{x} ( x ^{2} + 2 x ) ^{3}}{( 3 x ^{2} + x ) ^{2}} : \frac{e ^{x} x ( x + 2 ) ^{2} ( 3 x ^{3} + 19 x ^{2} + 13 x + 4 )}{( 3 x + 1 ) ^{2}}

= \frac{e ^{x} x ( x + 2 ) ^{2} ( 3 x ^{3} + 19 x ^{2} + 13 x + 4 )}{( 3 x + 1 ) ^{2}}

y = \frac{4}{x ^{2} + 4}, [- 10, 10]

\frac{d}{d x} ((x - y)^{2})

4 x - 17

\int \frac{cos ^{2} ( 4 x )}{4} d x

\frac{1}{s} \cdot (\frac{10}{s ^{2} + s + 10})

What is the derivative of (e^x(x^2+2x^3)/(3x^2+x)) ?
The derivative of (e^x(x^2+2x^3)/(3x^2+x)) is (e^xx(x+2)^2(3x^3+19x^2+13x+4))/((3x+1)^2)
What is the first derivative of (e^x(x^2+2x^3)/(3x^2+x)) ?
The first derivative of (e^x(x^2+2x^3)/(3x^2+x)) is (e^xx(x+2)^2(3x^3+19x^2+13x+4))/((3x+1)^2)