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

The derivative of ((x^2+1^3)/((4x-5)^5)) is (2(x^2+1)^2(2x^2-15x-10))/((4x-5)^6)

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

The first derivative of ((x^2+1^3)/((4x-5)^5)) is (2(x^2+1)^2(2x^2-15x-10))/((4x-5)^6)

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

\frac{2 ( x ^{2} + 1 ) ^{2} ( 2 x ^{2} - 15 x - 10 )}{( 4 x - 5 ) ^{6}}

Solution steps

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

Apply the Quotient Rule:

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

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

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

\frac{d}{d x} ((4 x - 5)^{5}) = 20 (4 x - 5)^{4}

= \frac{6 x ( x ^{2} + 1 ) ^{2} ( 4 x - 5 ) ^{5} - 20 ( 4 x - 5 ) ^{4} ( x ^{2} + 1 ) ^{3}}{( ( 4 x - 5 ) ^{5} ) ^{2}}

Simplify

\frac{6 x ( x ^{2} + 1 ) ^{2} ( 4 x - 5 ) ^{5} - 20 ( 4 x - 5 ) ^{4} ( x ^{2} + 1 ) ^{3}}{( ( 4 x - 5 ) ^{5} ) ^{2}} : \frac{2 ( x ^{2} + 1 ) ^{2} ( 2 x ^{2} - 15 x - 10 )}{( 4 x - 5 ) ^{6}}

= \frac{2 ( x ^{2} + 1 ) ^{2} ( 2 x ^{2} - 15 x - 10 )}{( 4 x - 5 ) ^{6}}

What is the derivative of ((x^2+1^3)/((4x-5)^5)) ?
The derivative of ((x^2+1^3)/((4x-5)^5)) is (2(x^2+1)^2(2x^2-15x-10))/((4x-5)^6)
What is the first derivative of ((x^2+1^3)/((4x-5)^5)) ?
The first derivative of ((x^2+1^3)/((4x-5)^5)) is (2(x^2+1)^2(2x^2-15x-10))/((4x-5)^6)