Calculus Examples

$y = - 5 x - \frac{35}{x}$

Step 1

Write $y = - 5 x - \frac{35}{x}$ as a function.

$f (x) = - 5 x - \frac{35}{x}$

Step 2

Find the second derivative.

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Step 2.1

Find the first derivative.

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Step 2.1.1

By the Sum Rule, the derivative of $- 5 x - \frac{35}{x}$ with respect to $x$ is $\frac{d}{d x} [- 5 x] + \frac{d}{d x} [- \frac{35}{x}]$ .

$\frac{d}{d x} [- 5 x] + \frac{d}{d x} [- \frac{35}{x}]$

Step 2.1.2

Evaluate $\frac{d}{d x} [- 5 x]$ .

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Step 2.1.2.1

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5 x$ with respect to $x$ is $- 5 \frac{d}{d x} [x]$ .

$- 5 \frac{d}{d x} [x] + \frac{d}{d x} [- \frac{35}{x}]$

Step 2.1.2.2

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

$- 5 \cdot 1 + \frac{d}{d x} [- \frac{35}{x}]$

Step 2.1.2.3

Multiply $- 5$ by $1$ .

$- 5 + \frac{d}{d x} [- \frac{35}{x}]$

Step 2.1.3

Evaluate $\frac{d}{d x} [- \frac{35}{x}]$ .

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Step 2.1.3.1

Since $- 35$ is constant with respect to $x$ , the derivative of $- \frac{35}{x}$ with respect to $x$ is $- 35 \frac{d}{d x} [\frac{1}{x}]$ .

$- 5 - 35 \frac{d}{d x} [\frac{1}{x}]$

Step 2.1.3.2

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

$- 5 - 35 \frac{d}{d x} [x^{- 1}]$

Step 2.1.3.3

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

$- 5 - 35 (- x^{- 2})$

Step 2.1.3.4

Multiply $- 1$ by $- 35$ .

$- 5 + 35 x^{- 2}$

Step 2.1.4

Simplify.

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Step 2.1.4.1

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

$- 5 + 35 \frac{1}{x^{2}}$

Step 2.1.4.2

Combine $35$ and $\frac{1}{x^{2}}$ .

$- 5 + \frac{35}{x^{2}}$

Step 2.1.4.3

Reorder terms.

$f' (x) = \frac{35}{x^{2}} - 5$

Step 2.2

Find the second derivative.

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Step 2.2.1

By the Sum Rule, the derivative of $\frac{35}{x^{2}} - 5$ with respect to $x$ is $\frac{d}{d x} [\frac{35}{x^{2}}] + \frac{d}{d x} [- 5]$ .

$\frac{d}{d x} [\frac{35}{x^{2}}] + \frac{d}{d x} [- 5]$

Step 2.2.2

Evaluate $\frac{d}{d x} [\frac{35}{x^{2}}]$ .

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Step 2.2.2.1

Since $35$ is constant with respect to $x$ , the derivative of $\frac{35}{x^{2}}$ with respect to $x$ is $35 \frac{d}{d x} [\frac{1}{x^{2}}]$ .

$35 \frac{d}{d x} [\frac{1}{x^{2}}] + \frac{d}{d x} [- 5]$

Step 2.2.2.2

Rewrite $\frac{1}{x^{2}}$ as ${(x^{2})}^{- 1}$ .

$35 \frac{d}{d x} [{(x^{2})}^{- 1}] + \frac{d}{d x} [- 5]$

Step 2.2.2.3

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 $g (x) = x^{2}$ .

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Step 2.2.2.3.1

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

$35 (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{2}]) + \frac{d}{d x} [- 5]$

Step 2.2.2.3.2

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

$35 (- u^{- 2} \frac{d}{d x} [x^{2}]) + \frac{d}{d x} [- 5]$

Step 2.2.2.3.3

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

$35 (- {(x^{2})}^{- 2} \frac{d}{d x} [x^{2}]) + \frac{d}{d x} [- 5]$

Step 2.2.2.4

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

$35 (- {(x^{2})}^{- 2} (2 x)) + \frac{d}{d x} [- 5]$

Step 2.2.2.5

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

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Step 2.2.2.5.1

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

$35 (- x^{2 \cdot - 2} (2 x)) + \frac{d}{d x} [- 5]$

Step 2.2.2.5.2

Multiply $2$ by $- 2$ .

$35 (- x^{- 4} (2 x)) + \frac{d}{d x} [- 5]$

Step 2.2.2.6

Multiply $2$ by $- 1$ .

$35 (- 2 x^{- 4} x) + \frac{d}{d x} [- 5]$

Step 2.2.2.7

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

$35 (- 2 (x^{1} x^{- 4})) + \frac{d}{d x} [- 5]$

Step 2.2.2.8

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

$35 (- 2 x^{1 - 4}) + \frac{d}{d x} [- 5]$

Step 2.2.2.9

Subtract $4$ from $1$ .

$35 (- 2 x^{- 3}) + \frac{d}{d x} [- 5]$

Step 2.2.2.10

Multiply $- 2$ by $35$ .

$- 70 x^{- 3} + \frac{d}{d x} [- 5]$

Step 2.2.3

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

$- 70 x^{- 3} + 0$

Step 2.2.4

Simplify.

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Step 2.2.4.1

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

$- 70 \frac{1}{x^{3}} + 0$

Step 2.2.4.2

Combine terms.

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Step 2.2.4.2.1

Combine $- 70$ and $\frac{1}{x^{3}}$ .

$\frac{- 70}{x^{3}} + 0$

Step 2.2.4.2.2

Move the negative in front of the fraction.

$- \frac{70}{x^{3}} + 0$

Step 2.2.4.2.3

Add $- \frac{70}{x^{3}}$ and $0$ .

$f'' (x) = - \frac{70}{x^{3}}$

Step 2.3

The second derivative of $f (x)$ with respect to $x$ is $- \frac{70}{x^{3}}$ .

$- \frac{70}{x^{3}}$

Step 3

Set the second derivative equal to $0$ then solve the equation $- \frac{70}{x^{3}} = 0$ .

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Step 3.1

Set the second derivative equal to $0$ .

$- \frac{70}{x^{3}} = 0$

Step 3.2

Set the numerator equal to zero.

$70 = 0$

Step 3.3

Since $70 \neq 0$ , there are no solutions.

No solution

Step 4

No values found that can make the second derivative equal to $0$ .

No Inflection Points