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 \frac{sin ( 5 x )}{5} d x

\frac{d}{d x} (- \frac{45}{8 ( 1 + x ) ^{\frac{7}{2}}})

y^{^{''}} - y^{^{'}} + y = 3 sin (t) - 27 e^{2 t}

\frac{1}{( s + 1 ) ( s - 1 )}

\frac{d}{d x} (a r (x) cot (\frac{x}{y}))

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}}