derivative of (2x/(sqrt(x))) — step-by-step solution

Let's perform the differentiation:

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

ⓘ The original denominator must remain nonzero. The fraction is simplified by dividing the numerator and the denominator by their common factor:

\frac{2 x}{\sqrt{x}}

2 x^{1 - \frac{1}{2}}

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

ⓘ To add fractions, put them over the same denominator, then add the numerators:

1 - \frac{1}{2}

\frac{2 - 1}{2}

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

ⓘ The numerator and the denominator are computed before the division:

2 - 1

1

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

ⓘ A constant factor stays in front of the derivative, (k*u)' = k*u':

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

2 \cdot \frac{d}{d x} (\sqrt{x})

■The derivative of sqrt(x) is 1/(2*sqrt(x)):

\frac{d}{d x} (\sqrt{x})

\frac{1}{2 \sqrt{x}}

2 \cdot \frac{1}{2 \sqrt{x}}

ⓘ Multiplying a fraction by a number multiplies its numerator:

2 \cdot \frac{1}{2 \sqrt{x}}

\frac{2 \cdot 1}{2 \sqrt{x}}

\frac{2}{2 \sqrt{x}}

■A fraction is simplified by dividing the numerator and the denominator by the same number:

\frac{2}{2 \sqrt{x}}

\frac{1}{\sqrt{x}}

▶Derivative:

\frac{1}{\sqrt{x}}