What is the derivative of x^{2/3}(6-x^{1/3}) ?

The derivative of x^{2/3}(6-x^{1/3}) is (4-x)/(x^{1/3)(-x+6)^{2/3}}

What is the first derivative of x^{2/3}(6-x^{1/3}) ?

The first derivative of x^{2/3}(6-x^{1/3}) is (4-x)/(x^{1/3)(-x+6)^{2/3}}

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

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

Solution steps

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

Apply the Product Rule:

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

f = x^{\frac{2}{3}}, g = (6 - x)^{\frac{1}{3}}

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

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

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

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

Simplify

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

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

\int_{8}^{13} \frac{y}{y ^{2} - 4 y - 5} d y

y^{^{''}} - 3 y^{^{'}} + 2 y = 4 e^{2 t}

\frac{d}{d x} (- 3 x + 7)

\frac{\partial}{\partial x} (3 e^{x y + 4})

\frac{d y}{d x} = x y + 5 x + 2 y + 10

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