derivative of (9x^2+9sqrt(x))/(8x)

1) Let's differentiate:

\frac{d}{d x} (\frac{9 x^{2} + 9 \sqrt{x}}{8 x})

ⓘ What the terms have in common is factored out, so that the fraction can be simplified:

\frac{9 x^{2} + 9 \sqrt{x}}{8 x}

\frac{9 \sqrt{x} \cdot (x^{\frac{3}{2}} + 1)}{8 x}

\frac{d}{d x} (\frac{9 \sqrt{x} \cdot (x^{\frac{3}{2}} + 1)}{8 x})

The fraction:

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

simplifies by the common factor:

\sqrt{x}

into the new quotient:

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

\frac{d}{d x} (\frac{9 \cdot (x^{\frac{3}{2}} + 1)}{8 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} (\frac{9 \cdot (x^{\frac{3}{2}} + 1)}{8 x^{\frac{2 - 1}{2}}})

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

2 - 1

1

\frac{d}{d x} (\frac{9 \cdot (x^{\frac{3}{2}} + 1)}{8 \sqrt{x}})

For

p1

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

and

q2

8 \sqrt{x}

, 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} (9 \cdot (x^{\frac{3}{2}} + 1))

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

\frac{d}{d x} (9 \cdot (x^{\frac{3}{2}} + 1))

9 \cdot \frac{d}{d x} (x^{\frac{3}{2}} + 1)

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

\frac{d}{d x} (x^{\frac{3}{2}} + 1)

\frac{d}{d x} (x^{\frac{3}{2}}) + 0

9 \cdot (\frac{d}{d x} (x^{\frac{3}{2}}) + 0)

ⓘ Adding 0 changes nothing:

\frac{d}{d x} (x^{\frac{3}{2}}) + 0

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

■The derivative of x^n is n*x^(n-1):

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

\frac{3}{2} \cdot \sqrt{x}

9 \cdot \frac{3}{2} \cdot \sqrt{x}

ⅰ

9 \cdot \frac{3}{2}

\frac{27}{2}

\frac{27}{2} \cdot \sqrt{x}

ⓘ Multiplying a fraction by a term multiplies its numerator:

\frac{27}{2} \cdot \sqrt{x}

\frac{27 \sqrt{x}}{2}

\frac{27 \sqrt{x}}{2}

3) Let's perform the differentiation

q2

\frac{d}{d x} (8 \sqrt{x})

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

\frac{d}{d x} (8 \sqrt{x})

8 \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}}

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

ⓘ Multiplying a fraction by a number multiplies its numerator:

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

\frac{8 \cdot 1}{2 \sqrt{x}}

\frac{8}{2 \sqrt{x}}

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

\frac{8}{2 \sqrt{x}}

\frac{4}{\sqrt{x}}

\frac{4}{\sqrt{x}}

4) Let's evaluate the resulting derivative (

\frac{p1}{q2}

)':

\frac{\frac{27 \sqrt{x}}{2} \cdot 8 \sqrt{x} - 9 \cdot (x^{\frac{3}{2}} + 1) \cdot \frac{4}{\sqrt{x}}}{{(8 \sqrt{x})}^{2}}

ⓘ Multiplying a fraction by a term multiplies its numerator:

\frac{27 \sqrt{x}}{2} \cdot 8 \sqrt{x}

\frac{27 \cdot \sqrt{x} \cdot 8 \cdot \sqrt{x}}{2}

\frac{216 x}{2}

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

\frac{9 \cdot (x^{\frac{3}{2}} + 1) \cdot 4}{\sqrt{x}}

\frac{36 \cdot (x^{\frac{3}{2}} + 1)}{\sqrt{x}}

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

\frac{216 x}{2} - \frac{36 \cdot (x^{\frac{3}{2}} + 1)}{\sqrt{x}}

\frac{216 x \cdot \sqrt{x} - 36 \cdot (x^{\frac{3}{2}} + 1) \cdot 2}{2 \sqrt{x}}

\frac{\frac{216 x \cdot \sqrt{x} - 36 \cdot (x^{\frac{3}{2}} + 1) \cdot 2}{2 \sqrt{x}}}{{(8 \sqrt{x})}^{2}}

ⓘ Multiplying powers with the same base adds the exponents:

x \sqrt{x}

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

■Multiplications are done before additions and subtractions:

36 \cdot 2

(30 + 6) \cdot 2

30 \cdot 2 + 6 \cdot 2

60 + 12

72

\frac{\frac{216 x^{1 + \frac{1}{2}} - 72 \cdot (x^{\frac{3}{2}} + 1)}{2 \sqrt{x}}}{{(8 \sqrt{x})}^{2}}

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

1 + \frac{1}{2}

\frac{2 + 1}{2}

\frac{\frac{216 x^{\frac{2 + 1}{2}} - 72 \cdot (x^{\frac{3}{2}} + 1)}{2 \sqrt{x}}}{{(8 \sqrt{x})}^{2}}

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

2 + 1

3

\frac{\frac{216 x^{\frac{3}{2}} - 72 \cdot (x^{\frac{3}{2}} + 1)}{2 \sqrt{x}}}{{(8 \sqrt{x})}^{2}}

ⓘ Dividing a fraction by

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

is multiplying its denominator by

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

\frac{\frac{216 x^{\frac{3}{2}} - 72 \cdot (x^{\frac{3}{2}} + 1)}{2 \sqrt{x}}}{{(8 \sqrt{x})}^{2}}

\frac{216 x^{\frac{3}{2}} - 72 \cdot (x^{\frac{3}{2}} + 1)}{2 \sqrt{x} \cdot {(8 \sqrt{x})}^{2}}

\frac{216 x^{\frac{3}{2}} - 72 \cdot (x^{\frac{3}{2}} + 1)}{2 \sqrt{x} \cdot {(8 \sqrt{x})}^{2}}

ⓘ Expand by multiplying each term inside the parentheses:

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

- (72 x^{\frac{3}{2}} + 72)

■The power of a product is the product of the powers:

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

8^{2} \cdot {(\sqrt{x})}^{2}

\frac{216 x^{\frac{3}{2}} - (72 x^{\frac{3}{2}} + 72)}{2 \sqrt{x} \cdot 8^{2} \cdot {(\sqrt{x})}^{2}}

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

- (72 x^{\frac{3}{2}} + 72)

- 72 x^{\frac{3}{2}} - 72

■Powers are computed before multiplications, additions and subtractions. An integer power is a repeated product:

8^{2}

8 \cdot 8

64

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

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

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

\frac{216 x^{\frac{3}{2}} - 72 x^{\frac{3}{2}} - 72}{2 \sqrt{x} \cdot 64 x^{\frac{1}{2} \cdot 2}}

ⓘ Like terms are grouped by adding their coefficients:

216 x^{\frac{3}{2}} - 72 x^{\frac{3}{2}}

144 x^{\frac{3}{2}}

■Multiplying a fraction by a number multiplies its numerator:

\frac{1}{2} \cdot 2

\frac{1 \cdot 2}{2}

\frac{2}{2}

■Multiplying powers with the same base adds the exponents:

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

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

ⅰ

2 \cdot 64

2 \cdot (60 + 4)

2 \cdot 60 + 2 \cdot 4

120 + 8

128

\frac{144 x^{\frac{3}{2}} - 72}{128 x^{\frac{1}{2} + \frac{2}{2}}}

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

\frac{1}{2} + \frac{2}{2}

\frac{1 + 2}{2}

\frac{144 x^{\frac{3}{2}} - 72}{128 x^{\frac{1 + 2}{2}}}

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

1 + 2

3

\frac{144 x^{\frac{3}{2}} - 72}{128 x^{\frac{3}{2}}}

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

\frac{144 x^{\frac{3}{2}} - 72}{128 x^{\frac{3}{2}}}

\frac{9 \cdot (2 x^{\frac{3}{2}} - 1)}{16 x^{\frac{3}{2}}}

\frac{9 \cdot (2 x^{\frac{3}{2}} - 1)}{16 x^{\frac{3}{2}}}

5) You can then resume the derivation:

ⅰ

\frac{d}{d x} (\frac{9 \cdot (x^{\frac{3}{2}} + 1)}{8 \sqrt{x}})

\frac{9 \cdot (2 x^{\frac{3}{2}} - 1)}{16 x^{\frac{3}{2}}}

▶Derivative:

\frac{9 \cdot (2 x^{\frac{3}{2}} - 1)}{16 x^{\frac{3}{2}}}