What is the derivative of (x-3^{4/5}(x+1)^{1/5}) ?

The derivative of (x-3^{4/5}(x+1)^{1/5}) is (5x+1)/(5(x-3)^{1/5)(x+1)^{4/5}}

What is the first derivative of (x-3^{4/5}(x+1)^{1/5}) ?

The first derivative of (x-3^{4/5}(x+1)^{1/5}) is (5x+1)/(5(x-3)^{1/5)(x+1)^{4/5}}

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\frac{d}{d x} ((x - 3)^{\frac{4}{5}} (x + 1)^{\frac{1}{5}})

\frac{5 x + 1}{5 ( x - 3 ) ^{\frac{1}{5}} ( x + 1 ) ^{\frac{4}{5}}}

Solution steps

\frac{d}{d x} ((x - 3)^{\frac{4}{5}} (x + 1)^{\frac{1}{5}})

Apply the Product Rule:

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

f = (x - 3)^{\frac{4}{5}}, g = (x + 1)^{\frac{1}{5}}

= \frac{d}{d x} ((x - 3)^{\frac{4}{5}}) (x + 1)^{\frac{1}{5}} + \frac{d}{d x} ((x + 1)^{\frac{1}{5}}) (x - 3)^{\frac{4}{5}}

\frac{d}{d x} ((x - 3)^{\frac{4}{5}}) = \frac{4}{5 ( x - 3 ) ^{\frac{1}{5}}}

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

= \frac{4}{5 ( x - 3 ) ^{\frac{1}{5}}} (x + 1)^{\frac{1}{5}} + \frac{1}{5 ( x + 1 ) ^{\frac{4}{5}}} (x - 3)^{\frac{4}{5}}

Simplify

\frac{4}{5 ( x - 3 ) ^{\frac{1}{5}}} (x + 1)^{\frac{1}{5}} + \frac{1}{5 ( x + 1 ) ^{\frac{4}{5}}} (x - 3)^{\frac{4}{5}} : \frac{5 x + 1}{5 ( x - 3 ) ^{\frac{1}{5}} ( x + 1 ) ^{\frac{4}{5}}}

= \frac{5 x + 1}{5 ( x - 3 ) ^{\frac{1}{5}} ( x + 1 ) ^{\frac{4}{5}}}

y = 2 x^{- 3}

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

x \to 4 - lim (\frac{x - 10}{x - 4})

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

What is the derivative of (x-3^{4/5}(x+1)^{1/5}) ?
The derivative of (x-3^{4/5}(x+1)^{1/5}) is (5x+1)/(5(x-3)^{1/5)(x+1)^{4/5}}
What is the first derivative of (x-3^{4/5}(x+1)^{1/5}) ?
The first derivative of (x-3^{4/5}(x+1)^{1/5}) is (5x+1)/(5(x-3)^{1/5)(x+1)^{4/5}}