What is the derivative of ((3x-5^3)/((2x^2+1)^4)) ?

The derivative of ((3x-5^3)/((2x^2+1)^4)) is ((3x-5)^2(-30x^2+80x+9))/((2x^2+1)^5)

What is the first derivative of ((3x-5^3)/((2x^2+1)^4)) ?

The first derivative of ((3x-5^3)/((2x^2+1)^4)) is ((3x-5)^2(-30x^2+80x+9))/((2x^2+1)^5)

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\frac{d}{d x} (\frac{( 3 x - 5 ) ^{3}}{( 2 x ^{2} + 1 ) ^{4}})

\frac{( 3 x - 5 ) ^{2} ( - 30 x ^{2} + 80 x + 9 )}{( 2 x ^{2} + 1 ) ^{5}}

Solution steps

\frac{d}{d x} (\frac{( 3 x - 5 ) ^{3}}{( 2 x ^{2} + 1 ) ^{4}})

Apply the Quotient Rule:

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

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

\frac{d}{d x} ((3 x - 5)^{3}) = 9 (3 x - 5)^{2}

\frac{d}{d x} ((2 x^{2} + 1)^{4}) = 16 x (2 x^{2} + 1)^{3}

= \frac{9 ( 3 x - 5 ) ^{2} ( 2 x ^{2} + 1 ) ^{4} - 16 x ( 2 x ^{2} + 1 ) ^{3} ( 3 x - 5 ) ^{3}}{( ( 2 x ^{2} + 1 ) ^{4} ) ^{2}}

Simplify

\frac{9 ( 3 x - 5 ) ^{2} ( 2 x ^{2} + 1 ) ^{4} - 16 x ( 2 x ^{2} + 1 ) ^{3} ( 3 x - 5 ) ^{3}}{( ( 2 x ^{2} + 1 ) ^{4} ) ^{2}} : \frac{( 3 x - 5 ) ^{2} ( - 30 x ^{2} + 80 x + 9 )}{( 2 x ^{2} + 1 ) ^{5}}

= \frac{( 3 x - 5 ) ^{2} ( - 30 x ^{2} + 80 x + 9 )}{( 2 x ^{2} + 1 ) ^{5}}

\frac{\partial}{\partial y} (y)

y^{^{''}} - 2 y^{^{'}} - 48 y = 0

\int \frac{1}{9 e ^{- 7 x} + e ^{7 x}} d x

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

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

What is the derivative of ((3x-5^3)/((2x^2+1)^4)) ?
The derivative of ((3x-5^3)/((2x^2+1)^4)) is ((3x-5)^2(-30x^2+80x+9))/((2x^2+1)^5)
What is the first derivative of ((3x-5^3)/((2x^2+1)^4)) ?
The first derivative of ((3x-5^3)/((2x^2+1)^4)) is ((3x-5)^2(-30x^2+80x+9))/((2x^2+1)^5)