Calculus Examples

$w = {(2 x - 7)}^{- 1} (x + 5)$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (w) = \frac{d}{d x} ({(2 x - 7)}^{- 1} (x + 5))$

Step 2

The derivative of $w$ with respect to $x$ is $w'$ .

$w'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = {(2 x - 7)}^{- 1}$ and $g (x) = x + 5$ .

${(2 x - 7)}^{- 1} \frac{d}{d x} [x + 5] + (x + 5) \frac{d}{d x} [{(2 x - 7)}^{- 1}]$

Step 3.2

Differentiate.

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Step 3.2.1

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

${(2 x - 7)}^{- 1} (\frac{d}{d x} [x] + \frac{d}{d x} [5]) + (x + 5) \frac{d}{d x} [{(2 x - 7)}^{- 1}]$

Step 3.2.2

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

${(2 x - 7)}^{- 1} (1 + \frac{d}{d x} [5]) + (x + 5) \frac{d}{d x} [{(2 x - 7)}^{- 1}]$

Step 3.2.3

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

${(2 x - 7)}^{- 1} (1 + 0) + (x + 5) \frac{d}{d x} [{(2 x - 7)}^{- 1}]$

Step 3.2.4

Simplify the expression.

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Step 3.2.4.1

Add $1$ and $0$ .

${(2 x - 7)}^{- 1} \cdot 1 + (x + 5) \frac{d}{d x} [{(2 x - 7)}^{- 1}]$

Step 3.2.4.2

Multiply ${(2 x - 7)}^{- 1}$ by $1$ .

${(2 x - 7)}^{- 1} + (x + 5) \frac{d}{d x} [{(2 x - 7)}^{- 1}]$

Step 3.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = 2 x - 7$ .

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Step 3.3.1

To apply the Chain Rule, set $u$ as $2 x - 7$ .

${(2 x - 7)}^{- 1} + (x + 5) (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [2 x - 7])$

Step 3.3.2

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

${(2 x - 7)}^{- 1} + (x + 5) (- u^{- 2} \frac{d}{d x} [2 x - 7])$

Step 3.3.3

Replace all occurrences of $u$ with $2 x - 7$ .

${(2 x - 7)}^{- 1} + (x + 5) (- {(2 x - 7)}^{- 2} \frac{d}{d x} [2 x - 7])$

Step 3.4

Differentiate.

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Step 3.4.1

By the Sum Rule, the derivative of $2 x - 7$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [- 7]$ .

${(2 x - 7)}^{- 1} + (x + 5) (- {(2 x - 7)}^{- 2} (\frac{d}{d x} [2 x] + \frac{d}{d x} [- 7]))$

Step 3.4.2

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

${(2 x - 7)}^{- 1} + (x + 5) (- {(2 x - 7)}^{- 2} (2 \frac{d}{d x} [x] + \frac{d}{d x} [- 7]))$

Step 3.4.3

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

${(2 x - 7)}^{- 1} + (x + 5) (- {(2 x - 7)}^{- 2} (2 \cdot 1 + \frac{d}{d x} [- 7]))$

Step 3.4.4

Multiply $2$ by $1$ .

${(2 x - 7)}^{- 1} + (x + 5) (- {(2 x - 7)}^{- 2} (2 + \frac{d}{d x} [- 7]))$

Step 3.4.5

Since $- 7$ is constant with respect to $x$ , the derivative of $- 7$ with respect to $x$ is $0$ .

${(2 x - 7)}^{- 1} + (x + 5) (- {(2 x - 7)}^{- 2} (2 + 0))$

Step 3.4.6

Simplify the expression.

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Step 3.4.6.1

Add $2$ and $0$ .

${(2 x - 7)}^{- 1} + (x + 5) (- {(2 x - 7)}^{- 2} \cdot 2)$

Step 3.4.6.2

Multiply $2$ by $- 1$ .

${(2 x - 7)}^{- 1} + (x + 5) (- 2 {(2 x - 7)}^{- 2})$

Step 3.5

Simplify.

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Step 3.5.1

Reorder terms.

$- 2 {(2 x - 7)}^{- 2} (x + 5) + {(2 x - 7)}^{- 1}$

Step 3.5.2

Simplify each term.

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Step 3.5.2.1

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

$- 2 \frac{1}{{(2 x - 7)}^{2}} (x + 5) + {(2 x - 7)}^{- 1}$

Step 3.5.2.2

Combine $- 2$ and $\frac{1}{{(2 x - 7)}^{2}}$ .

$\frac{- 2}{{(2 x - 7)}^{2}} (x + 5) + {(2 x - 7)}^{- 1}$

Step 3.5.2.3

Move the negative in front of the fraction.

$- \frac{2}{{(2 x - 7)}^{2}} (x + 5) + {(2 x - 7)}^{- 1}$

Step 3.5.2.4

Apply the distributive property.

$- \frac{2}{{(2 x - 7)}^{2}} x - \frac{2}{{(2 x - 7)}^{2}} \cdot 5 + {(2 x - 7)}^{- 1}$

Step 3.5.2.5

Combine $x$ and $\frac{2}{{(2 x - 7)}^{2}}$ .

$- \frac{x \cdot 2}{{(2 x - 7)}^{2}} - \frac{2}{{(2 x - 7)}^{2}} \cdot 5 + {(2 x - 7)}^{- 1}$

Step 3.5.2.6

Multiply $- \frac{2}{{(2 x - 7)}^{2}} \cdot 5$ .

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Step 3.5.2.6.1

Multiply $5$ by $- 1$ .

$- \frac{x \cdot 2}{{(2 x - 7)}^{2}} - 5 \frac{2}{{(2 x - 7)}^{2}} + {(2 x - 7)}^{- 1}$

Step 3.5.2.6.2

Combine $- 5$ and $\frac{2}{{(2 x - 7)}^{2}}$ .

$- \frac{x \cdot 2}{{(2 x - 7)}^{2}} + \frac{- 5 \cdot 2}{{(2 x - 7)}^{2}} + {(2 x - 7)}^{- 1}$

Step 3.5.2.6.3

Multiply $- 5$ by $2$ .

$- \frac{x \cdot 2}{{(2 x - 7)}^{2}} + \frac{- 10}{{(2 x - 7)}^{2}} + {(2 x - 7)}^{- 1}$

Step 3.5.2.7

Simplify each term.

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Step 3.5.2.7.1

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

$- \frac{2 \cdot x}{{(2 x - 7)}^{2}} + \frac{- 10}{{(2 x - 7)}^{2}} + {(2 x - 7)}^{- 1}$

Step 3.5.2.7.2

Move the negative in front of the fraction.

$- \frac{2 x}{{(2 x - 7)}^{2}} - \frac{10}{{(2 x - 7)}^{2}} + {(2 x - 7)}^{- 1}$

Step 3.5.2.8

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

$- \frac{2 x}{{(2 x - 7)}^{2}} - \frac{10}{{(2 x - 7)}^{2}} + \frac{1}{2 x - 7}$

Step 3.5.3

To write $\frac{1}{2 x - 7}$ as a fraction with a common denominator, multiply by $\frac{2 x - 7}{2 x - 7}$ .

$- \frac{2 x}{{(2 x - 7)}^{2}} + \frac{1}{2 x - 7} \cdot \frac{2 x - 7}{2 x - 7} - \frac{10}{{(2 x - 7)}^{2}}$

Step 3.5.4

Write each expression with a common denominator of ${(2 x - 7)}^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.5.4.1

Multiply $\frac{1}{2 x - 7}$ by $\frac{2 x - 7}{2 x - 7}$ .

$- \frac{2 x}{{(2 x - 7)}^{2}} + \frac{2 x - 7}{(2 x - 7) (2 x - 7)} - \frac{10}{{(2 x - 7)}^{2}}$

Step 3.5.4.2

Raise $2 x - 7$ to the power of $1$ .

$- \frac{2 x}{{(2 x - 7)}^{2}} + \frac{2 x - 7}{{(2 x - 7)}^{1} (2 x - 7)} - \frac{10}{{(2 x - 7)}^{2}}$

Step 3.5.4.3

Raise $2 x - 7$ to the power of $1$ .

$- \frac{2 x}{{(2 x - 7)}^{2}} + \frac{2 x - 7}{{(2 x - 7)}^{1} {(2 x - 7)}^{1}} - \frac{10}{{(2 x - 7)}^{2}}$

Step 3.5.4.4

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

$- \frac{2 x}{{(2 x - 7)}^{2}} + \frac{2 x - 7}{{(2 x - 7)}^{1 + 1}} - \frac{10}{{(2 x - 7)}^{2}}$

Step 3.5.4.5

Add $1$ and $1$ .

$- \frac{2 x}{{(2 x - 7)}^{2}} + \frac{2 x - 7}{{(2 x - 7)}^{2}} - \frac{10}{{(2 x - 7)}^{2}}$

Step 3.5.5

Combine the numerators over the common denominator.

$\frac{- 2 x + 2 x - 7}{{(2 x - 7)}^{2}} - \frac{10}{{(2 x - 7)}^{2}}$

Step 3.5.6

Combine the opposite terms in $\frac{- 2 x + 2 x - 7}{{(2 x - 7)}^{2}} - \frac{10}{{(2 x - 7)}^{2}}$ .

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Step 3.5.6.1

Add $- 2 x$ and $2 x$ .

$\frac{0 - 7}{{(2 x - 7)}^{2}} - \frac{10}{{(2 x - 7)}^{2}}$

Step 3.5.6.2

Subtract $7$ from $0$ .

$\frac{- 7}{{(2 x - 7)}^{2}} - \frac{10}{{(2 x - 7)}^{2}}$

Step 3.5.7

Combine the numerators over the common denominator.

$\frac{- 7 - 10}{{(2 x - 7)}^{2}}$

Step 3.5.8

Subtract $10$ from $- 7$ .

$\frac{- 17}{{(2 x - 7)}^{2}}$

Step 3.5.9

Move the negative in front of the fraction.

$- \frac{17}{{(2 x - 7)}^{2}}$

Step 4

Reform the equation by setting the left side equal to the right side.

$w' = - \frac{17}{{(2 x - 7)}^{2}}$

Step 5

Replace $w'$ with $\frac{d w}{d x}$ .

$\frac{d w}{d x} = - \frac{17}{{(2 x - 7)}^{2}}$