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

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

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

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

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derivative of

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

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

Solution steps

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

Apply the Product Rule:

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

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

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

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

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

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

Simplify

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

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

\int 5 x^{24} d x

\int 6 x^{\frac{2}{5}} d x

tan (5 x)

\frac{2}{4 ( s - 3 ) ^{2} + 64}

\int cos^{4} (5 x + 1) d x

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