derivative of y=(3x+4)/(sqrt(2x-1))

We save the definition:

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

1) Let's differentiate:

\frac{d}{d x} (\frac{3 x + 4}{\sqrt{2 x - 1}})

For

p1

3 x + 4

and

q2

\sqrt{2 x - 1}

, use the differentiation rule:

\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} (3 x + 4)

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

\frac{d}{d x} (3 x + 4)

\frac{d}{d x} (3 x) + 0

\frac{d}{d x} (3 x) + 0

ⓘ Adding 0 changes nothing:

\frac{d}{d x} (3 x) + 0

\frac{d}{d x} (3 x)

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

\frac{d}{d x} (3 x)

3

3

3) Let's perform the differentiation

q2

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

We differentiate using:

\frac{d}{d x} (\sqrt{f}) = \frac{d}{d x} (f) \cdot \frac{1}{2 \sqrt{f}}

(where f =

2 x - 1

)

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

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

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

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

ⓘ Multiplying a fraction by a term multiplies its numerator:

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

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

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

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

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

\frac{d}{d x} (2 x) + 0

\frac{\frac{d}{d x} (2 x) + 0}{2 \cdot \sqrt{2 x - 1}}

ⓘ Adding 0 changes nothing:

\frac{d}{d x} (2 x) + 0

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

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

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

2

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

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

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

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

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

4) Let's evaluate the resulting derivative (

\frac{p1}{q2}

)':

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

ⓘ Multiplying a fraction by a term multiplies its numerator:

(3 x + 4) \cdot \frac{1}{\sqrt{2 x - 1}}

\frac{(3 x + 4) \cdot 1}{\sqrt{2 x - 1}}

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

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

3 \cdot \sqrt{2 x - 1} - \frac{3 x + 4}{\sqrt{2 x - 1}}

\frac{3 \cdot \sqrt{2 x - 1} \cdot \sqrt{2 x - 1} - (3 x + 4)}{\sqrt{2 x - 1}}

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

{(\sqrt{2 x - 1})}^{2}

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

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

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

- (3 x + 4)

- 3 x - 4

■Multiplying powers with the same base adds the exponents:

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

2 x - 1

■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{3 \cdot (2 x - 1) - 3 x - 4}{\sqrt{2 x - 1}}}{{(2 x - 1)}^{1}}

\frac{\frac{3 \cdot (2 x - 1) - 3 x - 4}{\sqrt{2 x - 1}}}{2 x - 1}

ⓘ Dividing a fraction by

2 x - 1

is multiplying its denominator by

2 x - 1

\frac{\frac{3 \cdot (2 x - 1) - 3 x - 4}{\sqrt{2 x - 1}}}{2 x - 1}

\frac{3 \cdot (2 x - 1) - 3 x - 4}{\sqrt{2 x - 1} \cdot (2 x - 1)}

■Multiplying powers with the same base adds the exponents:

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

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

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

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

1 + 2

3

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

ⓘ Expand by multiplying each term inside the parentheses:

3 \cdot (2 x - 1)

6 x - 3

\frac{6 x - 3 - 3 x - 4}{{(2 x - 1)}^{\frac{3}{2}}}

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

- 3 - 4

-7

■Like terms are grouped by adding their coefficients:

6 x - 3 x

3 x

\frac{3 x - 7}{{(2 x - 1)}^{\frac{3}{2}}}

5) You can then resume the differentiation:

ⅰ

\frac{d}{d x} (\frac{3 x + 4}{\sqrt{2 x - 1}})

\frac{3 x - 7}{{(2 x - 1)}^{\frac{3}{2}}}

▶Derivative:

\frac{3 x - 7}{{(2 x - 1)}^{\frac{3}{2}}}