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

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

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

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

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

\frac{x - 1}{4 x ^{\frac{3}{4}}} + \frac{1}{2 x ^{\frac{1}{4}}}

Solution steps

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

Simplify

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

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

Apply the Product Rule:

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

f = x^{\frac{1}{4}}, g = x - 1

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

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

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

= \frac{1}{4 x ^{\frac{3}{4}}} (x - 1) + \frac{1}{2 x ^{\frac{1}{2}}} x^{\frac{1}{4}}

Simplify

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

= \frac{x - 1}{4 x ^{\frac{3}{4}}} + \frac{1}{2 x ^{\frac{1}{4}}}

x \to - 2 lim ((5 x + 7)^{4})

2^{x^{2} - 3}

\int_{- \infty}^{20} r e^{\frac{r}{5}} d r

\int \frac{2 x + 1}{x + 1} d x

\int - 1 ln (x^{2} - 1) d x

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