derivative of (4-2x-x^2/(x^2-6))

1) Let's perform the differentiation:

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

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

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

0 + \frac{d}{d x} (- 2 x) + \frac{d}{d x} (- \frac{x^{2}}{x^{2} - 6})

0 + \frac{d}{d x} (- 2 x) + \frac{d}{d x} (- \frac{x^{2}}{x^{2} - 6})

We differentiate

p1

x^{2}

and

q2

x^{2} - 6

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} (x^{2})

ⓘ The derivative of x^n is n*x^(n-1):

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

2 x

2 x

3) Let's differentiate

q2

\frac{d}{d x} (x^{2} - 6)

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

\frac{d}{d x} (x^{2} - 6)

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

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

ⓘ Adding 0 changes nothing:

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

\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 x

4) Let's evaluate the resulting derivative (

\frac{p1}{q2}

)':

- \frac{2 x \cdot (x^{2} - 6) - x^{2} \cdot 2 x}{{(x^{2} - 6)}^{2}}

ⓘ Multiplying powers with the same base adds the exponents:

x^{2} x

x^{3}

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

ⓘ The minus sign of the fraction goes into the numerator or the denominator, changing the sign of each of its terms:

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

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

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

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

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

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

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

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

- (x^{2} - 6)

- x^{2} + 6

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

ⓘ Like terms are grouped by adding their coefficients:

- x^{2} + x^{2}

0

\frac{2 x \cdot (0 + 6)}{{(x^{2} - 6)}^{2}}

ⓘ Adding 0 changes nothing:

0 + 6

6

ⅰ

2 \cdot 6

12

\frac{12 x}{{(x^{2} - 6)}^{2}}

5) Then you can resume the derivation:

ⅰ

\frac{d}{d x} (- \frac{x^{2}}{x^{2} - 6})

\frac{12 x}{{(x^{2} - 6)}^{2}}

- 2 + \frac{12 x}{{(x^{2} - 6)}^{2}}

▶Derivative:

\frac{12 x}{{(x^{2} - 6)}^{2}} - 2