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

The derivative of (4x^5/((2x^3-2)^3)) is (x^4(-4x^3-5))/(2(x^3-1)^4)

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

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

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

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

Solution steps

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

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

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

Apply the Quotient Rule:

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

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

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

\frac{d}{d x} ((2 x^{3} - 2)^{3}) = 18 x^{2} (2 x^{3} - 2)^{2}

= 4 \cdot \frac{5 x ^{4} ( 2 x ^{3} - 2 ) ^{3} - 18 x ^{2} ( 2 x ^{3} - 2 ) ^{2} x ^{5}}{( ( 2 x ^{3} - 2 ) ^{3} ) ^{2}}

Simplify

4 \cdot \frac{5 x ^{4} ( 2 x ^{3} - 2 ) ^{3} - 18 x ^{2} ( 2 x ^{3} - 2 ) ^{2} x ^{5}}{( ( 2 x ^{3} - 2 ) ^{3} ) ^{2}} : \frac{x ^{4} ( - 4 x ^{3} - 5 )}{2 ( x ^{3} - 1 ) ^{4}}

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

x \to 0 - lim (\frac{1}{x} \cdot e^{\frac{1}{x}})

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

\frac{d}{d y} (x - 5 ln ∣ x ∣ + y x)

h \to 0 lim (3)

y = cot (x)

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