Calculus Examples

$- \frac{10 x}{{(x^{2} + 5)}^{2}}$

Step 1

Since $- 10$ is constant with respect to $x$ , the derivative of $- \frac{10 x}{{(x^{2} + 5)}^{2}}$ with respect to $x$ is $- 10 \frac{d}{d x} [\frac{x}{{(x^{2} + 5)}^{2}}]$ .

$- 10 \frac{d}{d x} [\frac{x}{{(x^{2} + 5)}^{2}}]$

Step 2

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = x$ and $g (x) = {(x^{2} + 5)}^{2}$ .

$- 10 \frac{{(x^{2} + 5)}^{2} \frac{d}{d x} [x] - x \frac{d}{d x} [{(x^{2} + 5)}^{2}]}{{({(x^{2} + 5)}^{2})}^{2}}$

Step 3

Differentiate using the Power Rule.

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Step 3.1

Multiply the exponents in ${({(x^{2} + 5)}^{2})}^{2}$ .

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Step 3.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- 10 \frac{{(x^{2} + 5)}^{2} \frac{d}{d x} [x] - x \frac{d}{d x} [{(x^{2} + 5)}^{2}]}{{(x^{2} + 5)}^{2 \cdot 2}}$

Step 3.1.2

Multiply $2$ by $2$ .

$- 10 \frac{{(x^{2} + 5)}^{2} \frac{d}{d x} [x] - x \frac{d}{d x} [{(x^{2} + 5)}^{2}]}{{(x^{2} + 5)}^{4}}$

Step 3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$- 10 \frac{{(x^{2} + 5)}^{2} \cdot 1 - x \frac{d}{d x} [{(x^{2} + 5)}^{2}]}{{(x^{2} + 5)}^{4}}$

Step 3.3

Multiply ${(x^{2} + 5)}^{2}$ by $1$ .

$- 10 \frac{{(x^{2} + 5)}^{2} - x \frac{d}{d x} [{(x^{2} + 5)}^{2}]}{{(x^{2} + 5)}^{4}}$

Step 4

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = x^{2} + 5$ .

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Step 4.1

To apply the Chain Rule, set $u$ as $x^{2} + 5$ .

$- 10 \frac{{(x^{2} + 5)}^{2} - x (\frac{d}{d u} [u^{2}] \frac{d}{d x} [x^{2} + 5])}{{(x^{2} + 5)}^{4}}$

Step 4.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 2$ .

$- 10 \frac{{(x^{2} + 5)}^{2} - x (2 u \frac{d}{d x} [x^{2} + 5])}{{(x^{2} + 5)}^{4}}$

Step 4.3

Replace all occurrences of $u$ with $x^{2} + 5$ .

$- 10 \frac{{(x^{2} + 5)}^{2} - x (2 (x^{2} + 5) \frac{d}{d x} [x^{2} + 5])}{{(x^{2} + 5)}^{4}}$

Step 5

Simplify with factoring out.

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Step 5.1

Multiply $2$ by $- 1$ .

$- 10 \frac{{(x^{2} + 5)}^{2} - 2 x ((x^{2} + 5) \frac{d}{d x} [x^{2} + 5])}{{(x^{2} + 5)}^{4}}$

Step 5.2

Factor $x^{2} + 5$ out of ${(x^{2} + 5)}^{2} - 2 x ((x^{2} + 5) \frac{d}{d x} [x^{2} + 5])$ .

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Step 5.2.1

Factor $x^{2} + 5$ out of ${(x^{2} + 5)}^{2}$ .

$- 10 \frac{(x^{2} + 5) (x^{2} + 5) - 2 x ((x^{2} + 5) \frac{d}{d x} [x^{2} + 5])}{{(x^{2} + 5)}^{4}}$

Step 5.2.2

Factor $x^{2} + 5$ out of $- 2 x ((x^{2} + 5) \frac{d}{d x} [x^{2} + 5])$ .

$- 10 \frac{(x^{2} + 5) (x^{2} + 5) + (x^{2} + 5) (- 2 x (\frac{d}{d x} [x^{2} + 5]))}{{(x^{2} + 5)}^{4}}$

Step 5.2.3

Factor $x^{2} + 5$ out of $(x^{2} + 5) (x^{2} + 5) + (x^{2} + 5) (- 2 x (\frac{d}{d x} [x^{2} + 5]))$ .

$- 10 \frac{(x^{2} + 5) (x^{2} + 5 - 2 x (\frac{d}{d x} [x^{2} + 5]))}{{(x^{2} + 5)}^{4}}$

Step 6

Cancel the common factors.

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Step 6.1

Factor $x^{2} + 5$ out of ${(x^{2} + 5)}^{4}$ .

$- 10 \frac{(x^{2} + 5) (x^{2} + 5 - 2 x (\frac{d}{d x} [x^{2} + 5]))}{(x^{2} + 5) {(x^{2} + 5)}^{3}}$

Step 6.2

Cancel the common factor.

$- 10 \frac{(x^{2} + 5) (x^{2} + 5 - 2 x (\frac{d}{d x} [x^{2} + 5]))}{(x^{2} + 5) {(x^{2} + 5)}^{3}}$

Step 6.3

Rewrite the expression.

$- 10 \frac{x^{2} + 5 - 2 x (\frac{d}{d x} [x^{2} + 5])}{{(x^{2} + 5)}^{3}}$

Step 7

By the Sum Rule, the derivative of $x^{2} + 5$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [5]$ .

$- 10 \frac{x^{2} + 5 - 2 x (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [5])}{{(x^{2} + 5)}^{3}}$

Step 8

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$- 10 \frac{x^{2} + 5 - 2 x (2 x + \frac{d}{d x} [5])}{{(x^{2} + 5)}^{3}}$

Step 9

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$- 10 \frac{x^{2} + 5 - 2 x (2 x + 0)}{{(x^{2} + 5)}^{3}}$

Step 10

Simplify the expression.

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Step 10.1

Add $2 x$ and $0$ .

$- 10 \frac{x^{2} + 5 - 2 x (2 x)}{{(x^{2} + 5)}^{3}}$

Step 10.2

Multiply $2$ by $- 2$ .

$- 10 \frac{x^{2} + 5 - 4 x \cdot x}{{(x^{2} + 5)}^{3}}$

Step 11

Raise $x$ to the power of $1$ .

$- 10 \frac{x^{2} + 5 - 4 (x^{1} x)}{{(x^{2} + 5)}^{3}}$

Step 12

Raise $x$ to the power of $1$ .

$- 10 \frac{x^{2} + 5 - 4 (x^{1} x^{1})}{{(x^{2} + 5)}^{3}}$

Step 13

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 10 \frac{x^{2} + 5 - 4 x^{1 + 1}}{{(x^{2} + 5)}^{3}}$

Step 14

Add $1$ and $1$ .

$- 10 \frac{x^{2} + 5 - 4 x^{2}}{{(x^{2} + 5)}^{3}}$

Step 15

Subtract $4 x^{2}$ from $x^{2}$ .

$- 10 \frac{- 3 x^{2} + 5}{{(x^{2} + 5)}^{3}}$

Step 16

Combine $- 10$ and $\frac{- 3 x^{2} + 5}{{(x^{2} + 5)}^{3}}$ .

$\frac{- 10 (- 3 x^{2} + 5)}{{(x^{2} + 5)}^{3}}$

Step 17

Move the negative in front of the fraction.

$- \frac{10 (- 3 x^{2} + 5)}{{(x^{2} + 5)}^{3}}$

Step 18

Simplify.

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Step 18.1

Apply the distributive property.

$- \frac{10 (- 3 x^{2}) + 10 \cdot 5}{{(x^{2} + 5)}^{3}}$

Step 18.2

Simplify each term.

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Step 18.2.1

Multiply $- 3$ by $10$ .

$- \frac{- 30 x^{2} + 10 \cdot 5}{{(x^{2} + 5)}^{3}}$

Step 18.2.2

Multiply $10$ by $5$ .

$- \frac{- 30 x^{2} + 50}{{(x^{2} + 5)}^{3}}$

Step 18.3

Factor $10$ out of $- 30 x^{2} + 50$ .

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Step 18.3.1

Factor $10$ out of $- 30 x^{2}$ .

$- \frac{10 (- 3 x^{2}) + 50}{{(x^{2} + 5)}^{3}}$

Step 18.3.2

Factor $10$ out of $50$ .

$- \frac{10 (- 3 x^{2}) + 10 (5)}{{(x^{2} + 5)}^{3}}$

Step 18.3.3

Factor $10$ out of $10 (- 3 x^{2}) + 10 (5)$ .

$- \frac{10 (- 3 x^{2} + 5)}{{(x^{2} + 5)}^{3}}$

Step 18.4

Factor $- 1$ out of $- 3 x^{2}$ .

$- \frac{10 (- (3 x^{2}) + 5)}{{(x^{2} + 5)}^{3}}$

Step 18.5

Rewrite $5$ as $- 1 (- 5)$ .

$- \frac{10 (- (3 x^{2}) - 1 (- 5))}{{(x^{2} + 5)}^{3}}$

Step 18.6

Factor $- 1$ out of $- (3 x^{2}) - 1 (- 5)$ .

$- \frac{10 (- (3 x^{2} - 5))}{{(x^{2} + 5)}^{3}}$

Step 18.7

Rewrite $- (3 x^{2} - 5)$ as $- 1 (3 x^{2} - 5)$ .

$- \frac{10 (- 1 (3 x^{2} - 5))}{{(x^{2} + 5)}^{3}}$

Step 18.8

Move the negative in front of the fraction.

$- - \frac{10 (3 x^{2} - 5)}{{(x^{2} + 5)}^{3}}$

Step 18.9

Multiply $- 1$ by $- 1$ .

$1 \frac{10 (3 x^{2} - 5)}{{(x^{2} + 5)}^{3}}$

Step 18.10

Multiply $\frac{10 (3 x^{2} - 5)}{{(x^{2} + 5)}^{3}}$ by $1$ .

$\frac{10 (3 x^{2} - 5)}{{(x^{2} + 5)}^{3}}$