derivative of y=(5x-2)/(sqrt(x))

We save the definition:

y = \frac{5 x - 2}{\sqrt{x}}

1) Let's perform the differentiation:

\frac{d}{d x} (\frac{5 x - 2}{\sqrt{x}})

We differentiate

p1

5 x - 2

and

q2

\sqrt{x}

using:

\frac{d}{d x} (\frac{p1}{q2}) = \frac{\frac{d}{d x} (p1) \cdot q2 - p1 \cdot \frac{d}{d x} (q2)}{{q2}^{2}}

2) Let's differentiate

p1

\frac{d}{d x} (5 x - 2)

ⓘ The derivative of a sum is the sum of the derivatives:

\frac{d}{d x} (5 x - 2)

\frac{d}{d x} (5 x) + 0

\frac{d}{d x} (5 x) + 0

ⓘ Adding 0 changes nothing:

\frac{d}{d x} (5 x) + 0

\frac{d}{d x} (5 x)

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

\frac{d}{d x} (5 x)

5

5

3) Let's differentiate

q2

\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}}

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

4) Let's evaluate the resulting derivative (

\frac{p1}{q2}

)':

\frac{5 \sqrt{x} - (5 x - 2) \cdot \frac{1}{2 \sqrt{x}}}{{(\sqrt{x})}^{2}}

ⓘ Multiplying a fraction by a term multiplies its numerator:

(5 x - 2) \cdot \frac{1}{2 \sqrt{x}}

\frac{(5 x - 2) \cdot 1}{2 \sqrt{x}}

\frac{5 x - 2}{2 \sqrt{x}}

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

5 \sqrt{x} - \frac{5 x - 2}{2 \sqrt{x}}

\frac{5 \sqrt{x} \cdot 2 \sqrt{x} - (5 x - 2)}{2 \sqrt{x}}

■A power of a power multiplies the exponents, (a^n)^m = a^(n*m):

{(\sqrt{x})}^{2}

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

\frac{\frac{5 \sqrt{x} \cdot 2 \sqrt{x} - (5 x - 2)}{2 \sqrt{x}}}{x^{\frac{1}{2} \cdot 2}}

ⓘ A minus sign before a parenthesis changes the sign of each of its terms:

- (5 x - 2)

- 5 x + 2

■Multiplications are done before additions and subtractions:

5 \cdot 2

10

■Multiplying powers with the same base adds the exponents:

\sqrt{x} \cdot \sqrt{x}

x

■Multiplying a fraction by a number multiplies its numerator:

\frac{1}{2} \cdot 2

\frac{1 \cdot 2}{2}

\frac{2}{2}

■We simplify the fraction by dividing the numerator and the denominator by

2

, their greatest common divisor:

\frac{2}{2}

1

\frac{\frac{10 x - 5 x + 2}{2 \sqrt{x}}}{x^{1}}

ⓘ Like terms are grouped by adding their coefficients:

10 x - 5 x

5 x

\frac{\frac{5 x + 2}{2 \sqrt{x}}}{x}

ⓘ Dividing a fraction by

x

is multiplying its denominator by

x

\frac{\frac{5 x + 2}{2 \sqrt{x}}}{x}

\frac{5 x + 2}{2 \sqrt{x} \cdot x}

\frac{5 x + 2}{2 \sqrt{x} \cdot x}

ⓘ Multiplying powers with the same base adds the exponents:

\sqrt{x} x

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

\frac{5 x + 2}{2 x^{\frac{1}{2} + 1}}

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

\frac{1}{2} + 1

\frac{1 + 2}{2}

\frac{5 x + 2}{2 x^{\frac{1 + 2}{2}}}

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

1 + 2

3

\frac{5 x + 2}{2 x^{\frac{3}{2}}}

5) Then you can continue the derivation:

ⅰ

\frac{d}{d x} (\frac{5 x - 2}{\sqrt{x}})

\frac{5 x + 2}{2 x^{\frac{3}{2}}}

▶Derivative:

\frac{5 x + 2}{2 x^{\frac{3}{2}}}