derivative of f(x)=((sqrt(x)-2))/((2x^2+3))

We save the definition:

f (x) = \frac{\sqrt{x} - 2}{2 x^{2} + 3}

1) Let's perform the differentiation:

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

For

p1

\sqrt{x} - 2

and

q2

2 x^{2} + 3

, 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} (\sqrt{x} - 2)

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

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

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

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

ⓘ Adding 0 changes nothing:

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

\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}}

3) Let's differentiate

q2

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

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

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

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

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

ⓘ Adding 0 changes nothing:

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

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

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

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

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

■The derivative of x^n is n*x^(n-1):

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

2 x

2 \cdot 2 x

ⅰ

2 \cdot 2

4

4 x

4) We evaluate the resulting derivative (

\frac{p1}{q2}

)':

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

ⓘ Multiplying a fraction by a term multiplies its numerator:

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

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

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

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

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

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

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

ⓘ Multiplications are done before additions and subtractions:

4 \cdot 2

8

■Multiplying powers with the same base adds the exponents:

x \sqrt{x}

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

■The minus sign is placed in front of the product:

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

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

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

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

1 + \frac{1}{2}

\frac{2 + 1}{2}

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

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

2 + 1

3

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

ⓘ Dividing a fraction by

{(2 x^{2} + 3)}^{2}

is multiplying its denominator by

{(2 x^{2} + 3)}^{2}

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

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

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

ⓘ Expand by multiplying each term inside the parentheses:

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

8 \sqrt{x} - 16

\frac{2 x^{2} + 3 - (8 \sqrt{x} - 16) \cdot x^{\frac{3}{2}}}{2 \sqrt{x} \cdot {(2 x^{2} + 3)}^{2}}

ⓘ Expand by multiplying each term inside the parentheses:

- (8 \sqrt{x} - 16) \cdot x^{\frac{3}{2}}

- (8 x^{\frac{1}{2} + \frac{3}{2}} - 16 x^{\frac{3}{2}})

\frac{2 x^{2} + 3 - (8 x^{\frac{1}{2} + \frac{3}{2}} - 16 x^{\frac{3}{2}})}{2 \sqrt{x} \cdot {(2 x^{2} + 3)}^{2}}

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

- (8 x^{\frac{1}{2} + \frac{3}{2}} - 16 x^{\frac{3}{2}})

- 8 x^{\frac{1}{2} + \frac{3}{2}} + 16 x^{\frac{3}{2}}

\frac{2 x^{2} + 3 - 8 x^{\frac{1}{2} + \frac{3}{2}} + 16 x^{\frac{3}{2}}}{2 \sqrt{x} \cdot {(2 x^{2} + 3)}^{2}}

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

\frac{1}{2} + \frac{3}{2}

\frac{1 + 3}{2}

\frac{2 x^{2} + 3 - 8 x^{\frac{1 + 3}{2}} + 16 x^{\frac{3}{2}}}{2 \sqrt{x} \cdot {(2 x^{2} + 3)}^{2}}

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

1 + 3

4

\frac{2 x^{2} + 3 - 8 x^{\frac{4}{2}} + 16 x^{\frac{3}{2}}}{2 \sqrt{x} \cdot {(2 x^{2} + 3)}^{2}}

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

2

, their greatest common divisor:

\frac{4}{2}

2

\frac{2 x^{2} + 3 - 8 x^{2} + 16 x^{\frac{3}{2}}}{2 \sqrt{x} \cdot {(2 x^{2} + 3)}^{2}}

ⓘ Like terms are grouped by adding their coefficients:

2 x^{2} - 8 x^{2}

-6 x^{2}

\frac{- 6 x^{2} + 3 + 16 x^{\frac{3}{2}}}{2 \sqrt{x} \cdot {(2 x^{2} + 3)}^{2}}

5) Then you can continue the differentiation:

ⅰ

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

\frac{- 6 x^{2} + 3 + 16 x^{\frac{3}{2}}}{2 \sqrt{x} \cdot {(2 x^{2} + 3)}^{2}}

\frac{- 6 x^{2} + 3 + 16 x^{\frac{3}{2}}}{2 \sqrt{x} \cdot {(2 x^{2} + 3)}^{2}}

▶Derivative:

\frac{- 6 x^{2} + 16 x^{\frac{3}{2}} + 3}{2 \sqrt{x} \cdot {(2 x^{2} + 3)}^{2}}