What is the derivative of (x^{3/5}sqrt(x^3+2x-6)/((4x-3)^8)) ?

The derivative of (x^{3/5}sqrt(x^3+2x-6)/((4x-3)^8)) is

What is the first derivative of (x^{3/5}sqrt(x^3+2x-6)/((4x-3)^8)) ?

The first derivative of (x^{3/5}sqrt(x^3+2x-6)/((4x-3)^8)) is

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\frac{d}{d x} (\frac{x ^{\frac{3}{5}} x ^{3} + 2 x - 6}{( 4 x - 3 ) ^{8}})

\frac{60 x ^{4} - 45 x ^{3} + 40 x ^{2} - 296 x x ^{3} + 2 x - 6 ( x ^{3} + 2 x - 6 ) ^{\frac{1}{2}} - 30 x - 18 ( x ^{3} + 2 x - 6 ) ^{\frac{1}{2}} x ^{3} + 2 x - 6}{10 x ^{\frac{2}{5}} ( 4 x - 3 ) ^{9} ( x ^{3} + 2 x - 6 ) ^{\frac{1}{2}}}

Solution steps

\frac{d}{d x} (\frac{x ^{\frac{3}{5}} x ^{3} + 2 x - 6}{( 4 x - 3 ) ^{8}})

Apply the Quotient Rule:

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

= \frac{\frac{d}{d x} ( x ^{\frac{3}{5}} x ^{3} + 2 x - 6 ) ( 4 x - 3 ) ^{8} - \frac{d}{d x} ( ( 4 x - 3 ) ^{8} ) x ^{\frac{3}{5}} x ^{3} + 2 x - 6}{( ( 4 x - 3 ) ^{8} ) ^{2}}

\frac{d}{d x} (x^{\frac{3}{5}} x^{3} + 2 x - 6) = \frac{6 x ^{3} + 2 x - 6 ( x ^{3} + 2 x - 6 ) ^{\frac{1}{2}} + 5 x ( 3 x ^{2} + 2 )}{10 x ^{\frac{2}{5}} ( x ^{3} + 2 x - 6 ) ^{\frac{1}{2}}}

\frac{d}{d x} ((4 x - 3)^{8}) = 32 (4 x - 3)^{7}

= \frac{\frac{6 x ^{3} + 2 x - 6 ( x ^{3} + 2 x - 6 ) ^{\frac{1}{2}} + 5 x ( 3 x ^{2} + 2 )}{10 x ^{\frac{2}{5}} ( x ^{3} + 2 x - 6 ) ^{\frac{1}{2}}} ( 4 x - 3 ) ^{8} - 32 ( 4 x - 3 ) ^{7} x ^{\frac{3}{5}} x ^{3} + 2 x - 6}{( ( 4 x - 3 ) ^{8} ) ^{2}}

Simplify

\frac{\frac{6 x ^{3} + 2 x - 6 ( x ^{3} + 2 x - 6 ) ^{\frac{1}{2}} + 5 x ( 3 x ^{2} + 2 )}{10 x ^{\frac{2}{5}} ( x ^{3} + 2 x - 6 ) ^{\frac{1}{2}}} ( 4 x - 3 ) ^{8} - 32 ( 4 x - 3 ) ^{7} x ^{\frac{3}{5}} x ^{3} + 2 x - 6}{( ( 4 x - 3 ) ^{8} ) ^{2}} : \frac{60 x ^{4} - 45 x ^{3} + 40 x ^{2} - 296 x x ^{3} + 2 x - 6 ( x ^{3} + 2 x - 6 ) ^{\frac{1}{2}} - 30 x - 18 ( x ^{3} + 2 x - 6 ) ^{\frac{1}{2}} x ^{3} + 2 x - 6}{10 x ^{\frac{2}{5}} ( 4 x - 3 ) ^{9} ( x ^{3} + 2 x - 6 ) ^{\frac{1}{2}}}

= \frac{60 x ^{4} - 45 x ^{3} + 40 x ^{2} - 296 x x ^{3} + 2 x - 6 ( x ^{3} + 2 x - 6 ) ^{\frac{1}{2}} - 30 x - 18 ( x ^{3} + 2 x - 6 ) ^{\frac{1}{2}} x ^{3} + 2 x - 6}{10 x ^{\frac{2}{5}} ( 4 x - 3 ) ^{9} ( x ^{3} + 2 x - 6 ) ^{\frac{1}{2}}}

f (x) = \frac{4}{x - 2}

x \to 4 - lim (e^{\frac{3}{4 - x}})

\int x^{5} (x - 9) d x

\frac{40 s + 20}{s ^{2} + 9 s + 14}

\frac{\partial}{\partial x} (x - a)

What is the derivative of (x^{3/5}sqrt(x^3+2x-6)/((4x-3)^8)) ?
The derivative of (x^{3/5}sqrt(x^3+2x-6)/((4x-3)^8)) is
What is the first derivative of (x^{3/5}sqrt(x^3+2x-6)/((4x-3)^8)) ?
The first derivative of (x^{3/5}sqrt(x^3+2x-6)/((4x-3)^8)) is