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

The derivative of (x+1^{2/3}(3x^2-2)^3) is (2(3x^2-2)^3+54x(3x^2-2)^2(x+1))/(3(x+1)^{1/3)}

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

The first derivative of (x+1^{2/3}(3x^2-2)^3) is (2(3x^2-2)^3+54x(3x^2-2)^2(x+1))/(3(x+1)^{1/3)}

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

\frac{2 ( 3 x ^{2} - 2 ) ^{3} + 54 x ( 3 x ^{2} - 2 ) ^{2} ( x + 1 )}{3 ( x + 1 ) ^{\frac{1}{3}}}

Solution steps

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

Apply the Product Rule:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = (x + 1)^{\frac{2}{3}}, g = (3 x^{2} - 2)^{3}

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

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

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

= \frac{2}{3 ( x + 1 ) ^{\frac{1}{3}}} (3 x^{2} - 2)^{3} + 18 x (3 x^{2} - 2)^{2} (x + 1)^{\frac{2}{3}}

Simplify

\frac{2}{3 ( x + 1 ) ^{\frac{1}{3}}} (3 x^{2} - 2)^{3} + 18 x (3 x^{2} - 2)^{2} (x + 1)^{\frac{2}{3}} : \frac{2 ( 3 x ^{2} - 2 ) ^{3} + 54 x ( 3 x ^{2} - 2 ) ^{2} ( x + 1 )}{3 ( x + 1 ) ^{\frac{1}{3}}}

= \frac{2 ( 3 x ^{2} - 2 ) ^{3} + 54 x ( 3 x ^{2} - 2 ) ^{2} ( x + 1 )}{3 ( x + 1 ) ^{\frac{1}{3}}}

What is the derivative of (x+1^{2/3}(3x^2-2)^3) ?
The derivative of (x+1^{2/3}(3x^2-2)^3) is (2(3x^2-2)^3+54x(3x^2-2)^2(x+1))/(3(x+1)^{1/3)}
What is the first derivative of (x+1^{2/3}(3x^2-2)^3) ?
The first derivative of (x+1^{2/3}(3x^2-2)^3) is (2(3x^2-2)^3+54x(3x^2-2)^2(x+1))/(3(x+1)^{1/3)}