Calculus Examples

$f (x) = \frac{x}{x + \frac{c}{x}}$

Step 1

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 $g (x) = x + \frac{c}{x}$ .

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

Step 2

Differentiate using the Power Rule.

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Step 2.1

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

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

Step 2.2

Multiply $x + \frac{c}{x}$ by $1$ .

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

Step 3

Multiply $\frac{x + \frac{c}{x} - x \frac{d}{d x} [x + \frac{c}{x}]}{{(x + \frac{c}{x})}^{2}}$ by $\frac{x}{x}$ .

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

Step 4

Combine.

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

Step 5

Apply the distributive property.

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

Step 6

Cancel the common factor of $x$ .

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Step 6.1

Cancel the common factor.

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

Step 6.2

Rewrite the expression.

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

Step 7

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

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

Step 8

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

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

Step 9

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

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

Step 10

Add $1$ and $1$ .

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

Step 11

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

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

Step 12

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

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

Step 13

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

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

Step 14

Add $1$ and $1$ .

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

Step 15

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

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

Step 16

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

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

Step 17

Since $c$ is constant with respect to $x$ , the derivative of $\frac{c}{x}$ with respect to $x$ is $c \frac{d}{d x} [\frac{1}{x}]$ .

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

Step 18

Rewrite $\frac{1}{x}$ as $x^{- 1}$ .

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

Step 19

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

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

Step 20

Simplify.

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Step 20.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 20.2

Apply the distributive property.

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

Step 20.3

Simplify the numerator.

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Step 20.3.1

Simplify each term.

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Step 20.3.1.1

Multiply $- 1$ by $1$ .

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

Step 20.3.1.2

Rewrite using the commutative property of multiplication.

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

Step 20.3.1.3

Combine $\frac{1}{x^{2}}$ and $c$ .

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

Step 20.3.1.4

Cancel the common factor of $x^{2}$ .

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Step 20.3.1.4.1

Move the leading negative in $- \frac{c}{x^{2}}$ into the numerator.

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

Step 20.3.1.4.2

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

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

Step 20.3.1.4.3

Cancel the common factor.

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

Step 20.3.1.4.4

Rewrite the expression.

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

Step 20.3.1.5

Multiply $- 1$ by $- 1$ .

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

Step 20.3.1.6

Multiply $c$ by $1$ .

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

Step 20.3.2

Combine the opposite terms in $x^{2} + c - x^{2} + c$ .

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Step 20.3.2.1

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

$\frac{c + 0 + c}{x {(x + \frac{c}{x})}^{2}}$

Step 20.3.2.2

Add $c$ and $0$ .

$\frac{c + c}{x {(x + \frac{c}{x})}^{2}}$

Step 20.3.3

Add $c$ and $c$ .

$\frac{2 c}{x {(x + \frac{c}{x})}^{2}}$

Step 20.4

Simplify the denominator.

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Step 20.4.1

To write $x$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{2 c}{x {(\frac{x \cdot x}{x} + \frac{c}{x})}^{2}}$

Step 20.4.2

Combine the numerators over the common denominator.

$\frac{2 c}{x {(\frac{x \cdot x + c}{x})}^{2}}$

Step 20.4.3

Multiply $x$ by $x$ .

$\frac{2 c}{x {(\frac{x^{2} + c}{x})}^{2}}$

Step 20.4.4

Apply the product rule to $\frac{x^{2} + c}{x}$ .

$\frac{2 c}{x \frac{{(x^{2} + c)}^{2}}{x^{2}}}$

Step 20.5

Combine $x$ and $\frac{{(x^{2} + c)}^{2}}{x^{2}}$ .

$\frac{2 c}{\frac{x {(x^{2} + c)}^{2}}{x^{2}}}$

Step 20.6

Reduce the expression $\frac{x {(x^{2} + c)}^{2}}{x^{2}}$ by cancelling the common factors.

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Step 20.6.1

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

$\frac{2 c}{\frac{x ({(x^{2} + c)}^{2})}{x^{2}}}$

Step 20.6.2

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

$\frac{2 c}{\frac{x ({(x^{2} + c)}^{2})}{x \cdot x}}$

Step 20.6.3

Cancel the common factor.

$\frac{2 c}{\frac{x {(x^{2} + c)}^{2}}{x \cdot x}}$

Step 20.6.4

Rewrite the expression.

$\frac{2 c}{\frac{{(x^{2} + c)}^{2}}{x}}$

Step 20.7

Multiply the numerator by the reciprocal of the denominator.

$2 c \frac{x}{{(x^{2} + c)}^{2}}$

Step 20.8

Multiply $2 c \frac{x}{{(x^{2} + c)}^{2}}$ .

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Step 20.8.1

Combine $2$ and $\frac{x}{{(x^{2} + c)}^{2}}$ .

$c \frac{2 x}{{(x^{2} + c)}^{2}}$

Step 20.8.2

Combine $c$ and $\frac{2 x}{{(x^{2} + c)}^{2}}$ .

$\frac{c (2 x)}{{(x^{2} + c)}^{2}}$

Step 20.9

Move $2$ to the left of $c$ .

$\frac{2 c x}{{(x^{2} + c)}^{2}}$