What is the derivative of x(8-x^{1/3}) ?

The derivative of x(8-x^{1/3}) is (-4x+24)/(3(8-x)^{2/3)}

What is the first derivative of x(8-x^{1/3}) ?

The first derivative of x(8-x^{1/3}) is (-4x+24)/(3(8-x)^{2/3)}

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\frac{d}{d x} (x (8 - x)^{\frac{1}{3}})

\frac{- 4 x + 24}{3 ( 8 - x ) ^{\frac{2}{3}}}

Solution steps

\frac{d}{d x} (x (8 - x)^{\frac{1}{3}})

Apply the Product Rule:

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

f = x, g = (8 - x)^{\frac{1}{3}}

= \frac{d x}{d x} (8 - x)^{\frac{1}{3}} + \frac{d}{d x} ((8 - x)^{\frac{1}{3}}) x

\frac{d x}{d x} = 1

\frac{d}{d x} ((8 - x)^{\frac{1}{3}}) = - \frac{1}{3 ( 8 - x ) ^{\frac{2}{3}}}

= 1 \cdot (8 - x)^{\frac{1}{3}} + (- \frac{1}{3 ( 8 - x ) ^{\frac{2}{3}}}) x

Simplify

1 \cdot (8 - x)^{\frac{1}{3}} + (- \frac{1}{3 ( 8 - x ) ^{\frac{2}{3}}}) x : \frac{- 4 x + 24}{3 ( 8 - x ) ^{\frac{2}{3}}}

= \frac{- 4 x + 24}{3 ( 8 - x ) ^{\frac{2}{3}}}

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What is the derivative of x(8-x^{1/3}) ?
The derivative of x(8-x^{1/3}) is (-4x+24)/(3(8-x)^{2/3)}
What is the first derivative of x(8-x^{1/3}) ?
The first derivative of x(8-x^{1/3}) is (-4x+24)/(3(8-x)^{2/3)}