What is the derivative of ((2x^3+7(3x^2-5))/(x^2+1)) ?

The derivative of ((2x^3+7(3x^2-5))/(x^2+1)) is (18x^6+20x^4-30x^2+112x)/((x^2+1)^2)

What is the first derivative of ((2x^3+7(3x^2-5))/(x^2+1)) ?

The first derivative of ((2x^3+7(3x^2-5))/(x^2+1)) is (18x^6+20x^4-30x^2+112x)/((x^2+1)^2)

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\frac{d}{d x} (\frac{( 2 x ^{3} + 7 ) ( 3 x ^{2} - 5 )}{x ^{2} + 1})

\frac{18 x ^{6} + 20 x ^{4} - 30 x ^{2} + 112 x}{( x ^{2} + 1 ) ^{2}}

Solution steps

\frac{d}{d x} (\frac{( 2 x ^{3} + 7 ) ( 3 x ^{2} - 5 )}{x ^{2} + 1})

Apply the Quotient Rule:

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

= \frac{\frac{d}{d x} ( ( 2 x ^{3} + 7 ) ( 3 x ^{2} - 5 ) ) ( x ^{2} + 1 ) - \frac{d}{d x} ( x ^{2} + 1 ) ( 2 x ^{3} + 7 ) ( 3 x ^{2} - 5 )}{( x ^{2} + 1 ) ^{2}}

\frac{d}{d x} ((2 x^{3} + 7) (3 x^{2} - 5)) = 30 x^{4} - 30 x^{2} + 42 x

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

= \frac{( 30 x ^{4} - 30 x ^{2} + 42 x ) ( x ^{2} + 1 ) - 2 x ( 2 x ^{3} + 7 ) ( 3 x ^{2} - 5 )}{( x ^{2} + 1 ) ^{2}}

Expand

(30 x^{4} - 30 x^{2} + 42 x) (x^{2} + 1) - 2 x (2 x^{3} + 7) (3 x^{2} - 5) : 18 x^{6} + 20 x^{4} - 30 x^{2} + 112 x

= \frac{18 x ^{6} + 20 x ^{4} - 30 x ^{2} + 112 x}{( x ^{2} + 1 ) ^{2}}

x \to 7 lim (\frac{3}{( x - 7 ) ^{2}})

\frac{\partial}{\partial x} (x^{- 1} y^{2} + x y^{2} + 4 x y)

\int (1 - x) (6 + x^{2}) d x

sin (x \frac{π}{2})

\int e^{x} \cdot x^{3} d x

What is the derivative of ((2x^3+7(3x^2-5))/(x^2+1)) ?
The derivative of ((2x^3+7(3x^2-5))/(x^2+1)) is (18x^6+20x^4-30x^2+112x)/((x^2+1)^2)
What is the first derivative of ((2x^3+7(3x^2-5))/(x^2+1)) ?
The first derivative of ((2x^3+7(3x^2-5))/(x^2+1)) is (18x^6+20x^4-30x^2+112x)/((x^2+1)^2)