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

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

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

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

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

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

Solution steps

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

Apply the Product Rule:

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

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

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

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

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

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

Simplify

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

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

\frac{d y}{d x} + \frac{x ^{2}}{y} = 0

x \to 0 lim (x^{21 x})

\frac{d y}{d x} = \frac{x - y}{x + 2 y}

(9 + x) \frac{d y}{d x} = 6 + x

\frac{d}{d x} (\frac{x y}{2})

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