Calculus Examples

$\frac{50 x}{0.01 x^{2} + 1}$

Step 1

Since $50$ is constant with respect to $x$ , the derivative of $\frac{50 x}{0.01 x^{2} + 1}$ with respect to $x$ is $50 \frac{d}{d x} [\frac{x}{0.01 x^{2} + 1}]$ .

$50 \frac{d}{d x} [\frac{x}{0.01 x^{2} + 1}]$

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 $g (x) = 0.01 x^{2} + 1$ .

$50 \frac{(0.01 x^{2} + 1) \frac{d}{d x} [x] - x \frac{d}{d x} [0.01 x^{2} + 1]}{{(0.01 x^{2} + 1)}^{2}}$

Step 3

Differentiate.

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Step 3.1

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

$50 \frac{(0.01 x^{2} + 1) \cdot 1 - x \frac{d}{d x} [0.01 x^{2} + 1]}{{(0.01 x^{2} + 1)}^{2}}$

Step 3.2

Multiply $0.01 x^{2} + 1$ by $1$ .

$50 \frac{0.01 x^{2} + 1 - x \frac{d}{d x} [0.01 x^{2} + 1]}{{(0.01 x^{2} + 1)}^{2}}$

Step 3.3

By the Sum Rule, the derivative of $0.01 x^{2} + 1$ with respect to $x$ is $\frac{d}{d x} [0.01 x^{2}] + \frac{d}{d x} [1]$ .

$50 \frac{0.01 x^{2} + 1 - x (\frac{d}{d x} [0.01 x^{2}] + \frac{d}{d x} [1])}{{(0.01 x^{2} + 1)}^{2}}$

Step 3.4

Since $0.01$ is constant with respect to $x$ , the derivative of $0.01 x^{2}$ with respect to $x$ is $0.01 \frac{d}{d x} [x^{2}]$ .

$50 \frac{0.01 x^{2} + 1 - x (0.01 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [1])}{{(0.01 x^{2} + 1)}^{2}}$

Step 3.5

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

$50 \frac{0.01 x^{2} + 1 - x (0.01 (2 x) + \frac{d}{d x} [1])}{{(0.01 x^{2} + 1)}^{2}}$

Step 3.6

Multiply $2$ by $0.01$ .

$50 \frac{0.01 x^{2} + 1 - x (0.02 x + \frac{d}{d x} [1])}{{(0.01 x^{2} + 1)}^{2}}$

Step 3.7

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$50 \frac{0.01 x^{2} + 1 - x (0.02 x + 0)}{{(0.01 x^{2} + 1)}^{2}}$

Step 3.8

Simplify the expression.

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Step 3.8.1

Add $0.02 x$ and $0$ .

$50 \frac{0.01 x^{2} + 1 - x (0.02 x)}{{(0.01 x^{2} + 1)}^{2}}$

Step 3.8.2

Multiply $0.02$ by $- 1$ .

$50 \frac{0.01 x^{2} + 1 - 0.02 x \cdot x}{{(0.01 x^{2} + 1)}^{2}}$

Step 4

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

$50 \frac{0.01 x^{2} + 1 - 0.02 (x^{1} x)}{{(0.01 x^{2} + 1)}^{2}}$

Step 5

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

$50 \frac{0.01 x^{2} + 1 - 0.02 (x^{1} x^{1})}{{(0.01 x^{2} + 1)}^{2}}$

Step 6

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

$50 \frac{0.01 x^{2} + 1 - 0.02 x^{1 + 1}}{{(0.01 x^{2} + 1)}^{2}}$

Step 7

Add $1$ and $1$ .

$50 \frac{0.01 x^{2} + 1 - 0.02 x^{2}}{{(0.01 x^{2} + 1)}^{2}}$

Step 8

Subtract $0.02 x^{2}$ from $0.01 x^{2}$ .

$50 \frac{- 0.01 x^{2} + 1}{{(0.01 x^{2} + 1)}^{2}}$

Step 9

Combine $50$ and $\frac{- 0.01 x^{2} + 1}{{(0.01 x^{2} + 1)}^{2}}$ .

$\frac{50 (- 0.01 x^{2} + 1)}{{(0.01 x^{2} + 1)}^{2}}$

Step 10

Simplify.

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Step 10.1

Apply the distributive property.

$\frac{50 (- 0.01 x^{2}) + 50 \cdot 1}{{(0.01 x^{2} + 1)}^{2}}$

Step 10.2

Simplify each term.

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Step 10.2.1

Multiply $- 0.01$ by $50$ .

$\frac{- 0.5 x^{2} + 50 \cdot 1}{{(0.01 x^{2} + 1)}^{2}}$

Step 10.2.2

Multiply $50$ by $1$ .

$\frac{- 0.5 x^{2} + 50}{{(0.01 x^{2} + 1)}^{2}}$

Step 10.3

Simplify the numerator.

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Step 10.3.1

Factor $0.5$ out of $- 0.5 x^{2} + 50$ .

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Step 10.3.1.1

Factor $0.5$ out of $- 0.5 x^{2}$ .

$\frac{0.5 (- x^{2}) + 50}{{(0.01 x^{2} + 1)}^{2}}$

Step 10.3.1.2

Factor $0.5$ out of $50$ .

$\frac{0.5 (- x^{2}) + 0.5 (100)}{{(0.01 x^{2} + 1)}^{2}}$

Step 10.3.1.3

Factor $0.5$ out of $0.5 (- x^{2}) + 0.5 (100)$ .

$\frac{0.5 (- x^{2} + 100)}{{(0.01 x^{2} + 1)}^{2}}$

Step 10.3.2

Rewrite $100$ as $10^{2}$ .

$\frac{0.5 (- x^{2} + 10^{2})}{{(0.01 x^{2} + 1)}^{2}}$

Step 10.3.3

Reorder $- x^{2}$ and $10^{2}$ .

$\frac{0.5 (10^{2} - x^{2})}{{(0.01 x^{2} + 1)}^{2}}$

Step 10.3.4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 10$ and $b = x$ .

$\frac{0.5 (10 + x) (10 - x)}{{(0.01 x^{2} + 1)}^{2}}$

Step 10.4

Simplify the denominator.

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Step 10.4.1

Factor $0.01$ out of $0.01 x^{2} + 1$ .

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Step 10.4.1.1

Factor $0.01$ out of $0.01 x^{2}$ .

$\frac{0.5 (10 + x) (10 - x)}{{(0.01 (x^{2}) + 1)}^{2}}$

Step 10.4.1.2

Factor $0.01$ out of $1$ .

$\frac{0.5 (10 + x) (10 - x)}{{(0.01 x^{2} + 0.01 \cdot 100)}^{2}}$

Step 10.4.1.3

Factor $0.01$ out of $0.01 x^{2} + 0.01 \cdot 100$ .

$\frac{0.5 (10 + x) (10 - x)}{{(0.01 (x^{2} + 100))}^{2}}$

Step 10.4.2

Apply the product rule to $0.01 (x^{2} + 100)$ .

$\frac{0.5 (10 + x) (10 - x)}{{0.01}^{2} {(x^{2} + 100)}^{2}}$

Step 10.4.3

Raise $0.01$ to the power of $2$ .

$\frac{0.5 (10 + x) (10 - x)}{0.0001 {(x^{2} + 100)}^{2}}$

Step 10.5

Factor $0.5$ out of $0.5 (10 + x) (10 - x)$ .

$\frac{0.5 ((10 + x) (10 - x))}{0.0001 {(x^{2} + 100)}^{2}}$

Step 10.6

Factor $0.0001$ out of $0.0001 {(x^{2} + 100)}^{2}$ .

$\frac{0.5 ((10 + x) (10 - x))}{0.0001 ({(x^{2} + 100)}^{2})}$

Step 10.7

Separate fractions.

$\frac{0.5}{0.0001} \cdot \frac{(10 + x) (10 - x)}{{(x^{2} + 100)}^{2}}$

Step 10.8

Divide $0.5$ by $0.0001$ .

$5000 \frac{(10 + x) (10 - x)}{{(x^{2} + 100)}^{2}}$

Step 10.9

Combine $5000$ and $\frac{(10 + x) (10 - x)}{{(x^{2} + 100)}^{2}}$ .

$\frac{5000 (10 + x) (10 - x)}{{(x^{2} + 100)}^{2}}$