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

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

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

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

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

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

Solution steps

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

Apply the Product Rule:

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

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

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

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

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

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

Simplify

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

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

\int \frac{9}{( x - 1 ) ( x + 8 )} d x

4 x^{4} + 3 x^{3} + 2 x^{2} + x + 1

\frac{d}{d t} (6 ln (cos (t)))

\int \frac{( x + 4 )}{2 x + 3} d x

\int 5 e^{- 5 x} d x

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