What is the derivative of x^{1/5}(x+6) ?

The derivative of x^{1/5}(x+6) is (6x+6)/(5x^{4/5)}

What is the first derivative of x^{1/5}(x+6) ?

The first derivative of x^{1/5}(x+6) is (6x+6)/(5x^{4/5)}

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\frac{d}{d x} (x^{\frac{1}{5}} (x + 6))

\frac{6 x + 6}{5 x ^{\frac{4}{5}}}

Solution steps

\frac{d}{d x} (x^{\frac{1}{5}} (x + 6))

Apply the Product Rule:

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

f = x^{\frac{1}{5}}, g = x + 6

= \frac{d}{d x} (x^{\frac{1}{5}}) (x + 6) + \frac{d}{d x} (x + 6) x^{\frac{1}{5}}

\frac{d}{d x} (x^{\frac{1}{5}}) = \frac{1}{5 x ^{\frac{4}{5}}}

\frac{d}{d x} (x + 6) = 1

= \frac{1}{5 x ^{\frac{4}{5}}} (x + 6) + 1 \cdot x^{\frac{1}{5}}

Simplify

\frac{1}{5 x ^{\frac{4}{5}}} (x + 6) + 1 \cdot x^{\frac{1}{5}} : \frac{6 x + 6}{5 x ^{\frac{4}{5}}}

= \frac{6 x + 6}{5 x ^{\frac{4}{5}}}

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What is the derivative of x^{1/5}(x+6) ?
The derivative of x^{1/5}(x+6) is (6x+6)/(5x^{4/5)}
What is the first derivative of x^{1/5}(x+6) ?
The first derivative of x^{1/5}(x+6) is (6x+6)/(5x^{4/5)}