derivative of ((x^2-5)^3)/(x^2+2)

1) Let's differentiate:

\frac{d}{d x} (\frac{{(x^{2} - 5)}^{3}}{x^{2} + 2})

For

p1

{(x^{2} - 5)}^{3}

and

q2

x^{2} + 2

, 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 perform the differentiation

p1

\frac{d}{d x} ({(x^{2} - 5)}^{3})

We differentiate using:

\frac{d}{d x} (f^{n}) = n \cdot \frac{d}{d x} (f) \cdot f^{n - 1}

(where f =

x^{2} - 5

)

ⓘ The derivative of x^n is n*x^(n-1):

\frac{d}{d x} ({(x^{2} - 5)}^{3})

3 \cdot \frac{d}{d x} (x^{2} - 5) \cdot {(x^{2} - 5)}^{2}

3 \cdot \frac{d}{d x} (x^{2} - 5) \cdot {(x^{2} - 5)}^{2}

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

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

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

3 \cdot (\frac{d}{d x} (x^{2}) + 0) \cdot {(x^{2} - 5)}^{2}

ⓘ 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

3 \cdot 2 x \cdot {(x^{2} - 5)}^{2}

ⅰ

3 \cdot 2

6

6 x \cdot {(x^{2} - 5)}^{2}

3) Let's differentiate

q2

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

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

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

\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) We evaluate the resulting derivative (

\frac{p1}{q2}

)':

\frac{6 x \cdot {(x^{2} - 5)}^{2} \cdot (x^{2} + 2) - {(x^{2} - 5)}^{3} \cdot 2 x}{{(x^{2} + 2)}^{2}}

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

\frac{6 x \cdot {(x^{2} - 5)}^{2} \cdot (x^{2} + 2) - {(x^{2} - 5)}^{3} \cdot 2 x}{{(x^{2} + 2)}^{2}}

\frac{2 x \cdot (3 \cdot (x^{2} + 2) \cdot {(x^{2} - 5)}^{2} - {(x^{2} - 5)}^{3})}{{(x^{2} + 2)}^{2}}

\frac{2 x \cdot (3 \cdot (x^{2} + 2) \cdot {(x^{2} - 5)}^{2} - {(x^{2} - 5)}^{3})}{{(x^{2} + 2)}^{2}}

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

\frac{2 x \cdot (3 \cdot (x^{2} + 2) \cdot {(x^{2} - 5)}^{2} - {(x^{2} - 5)}^{3})}{{(x^{2} + 2)}^{2}}

\frac{2 x \cdot {(- (x^{2} - 5))}^{2} \cdot (3 \cdot (x^{2} + 2) - (x^{2} - 5))}{{(x^{2} + 2)}^{2}}

\frac{2 x \cdot {(- (x^{2} - 5))}^{2} \cdot (3 \cdot (x^{2} + 2) - (x^{2} - 5))}{{(x^{2} + 2)}^{2}}

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

- (x^{2} - 5)

- x^{2} + 5

\frac{2 x \cdot {(- x^{2} + 5)}^{2} \cdot (3 \cdot (x^{2} + 2) - x^{2} + 5)}{{(x^{2} + 2)}^{2}}

ⓘ Expand by multiplying each term inside the parentheses:

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

3 x^{2} + 6

\frac{2 x \cdot {(- x^{2} + 5)}^{2} \cdot (3 x^{2} + 6 - x^{2} + 5)}{{(x^{2} + 2)}^{2}}

ⓘ Parentheses are computed first:

6 + 5

11

■Like terms are grouped by adding their coefficients:

3 x^{2} - x^{2}

2 x^{2}

\frac{2 x \cdot {(- x^{2} + 5)}^{2} \cdot (2 x^{2} + 11)}{{(x^{2} + 2)}^{2}}

5) Then you can resume the differentiation:

ⅰ

\frac{d}{d x} (\frac{{(x^{2} - 5)}^{3}}{x^{2} + 2})

\frac{2 x \cdot {(- x^{2} + 5)}^{2} \cdot (2 x^{2} + 11)}{{(x^{2} + 2)}^{2}}

\frac{2 x \cdot {(- x^{2} + 5)}^{2} \cdot (2 x^{2} + 11)}{{(x^{2} + 2)}^{2}}

▶Derivative:

\frac{2 x \cdot (2 x^{2} + 11) \cdot {(5 - x^{2})}^{2}}{{(x^{2} + 2)}^{2}}