Calculus Examples

$f (x) = \frac{- 1}{x^{2}}$

Step 1

Find the second derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

Move the negative in front of the fraction.

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

Step 1.1.2

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 $f (x) = - 1$ and $g (x) = \frac{1}{x^{2}}$ .

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

Step 1.1.3

Differentiate.

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Step 1.1.3.1

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

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

Step 1.1.3.2

Multiply the exponents in ${(x^{2})}^{- 1}$ .

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Step 1.1.3.2.1

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

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

Step 1.1.3.2.2

Multiply $2$ by $- 1$ .

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

Step 1.1.3.3

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

$- (- 2 x^{- 3}) + \frac{1}{x^{2}} \frac{d}{d x} [- 1]$

Step 1.1.3.4

Multiply $- 2$ by $- 1$ .

$2 x^{- 3} + \frac{1}{x^{2}} \frac{d}{d x} [- 1]$

Step 1.1.3.5

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

$2 x^{- 3} + \frac{1}{x^{2}} \cdot 0$

Step 1.1.3.6

Simplify the expression.

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Step 1.1.3.6.1

Multiply $\frac{1}{x^{2}}$ by $0$ .

$2 x^{- 3} + 0$

Step 1.1.3.6.2

Add $2 x^{- 3}$ and $0$ .

$2 x^{- 3}$

Step 1.1.4

Simplify.

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Step 1.1.4.1

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

$2 \frac{1}{x^{3}}$

Step 1.1.4.2

Combine $2$ and $\frac{1}{x^{3}}$ .

$f' (x) = \frac{2}{x^{3}}$

Step 1.2

Find the second derivative.

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Step 1.2.1

Since $2$ is constant with respect to $x$ , the derivative of $\frac{2}{x^{3}}$ with respect to $x$ is $2 \frac{d}{d x} [\frac{1}{x^{3}}]$ .

$2 \frac{d}{d x} [\frac{1}{x^{3}}]$

Step 1.2.2

Apply basic rules of exponents.

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Step 1.2.2.1

Rewrite $\frac{1}{x^{3}}$ as ${(x^{3})}^{- 1}$ .

$2 \frac{d}{d x} [{(x^{3})}^{- 1}]$

Step 1.2.2.2

Multiply the exponents in ${(x^{3})}^{- 1}$ .

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Step 1.2.2.2.1

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

$2 \frac{d}{d x} [x^{3 \cdot - 1}]$

Step 1.2.2.2.2

Multiply $3$ by $- 1$ .

$2 \frac{d}{d x} [x^{- 3}]$

Step 1.2.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - 3$ .

$2 (- 3 x^{- 4})$

Step 1.2.4

Multiply $- 3$ by $2$ .

$- 6 x^{- 4}$

Step 1.2.5

Simplify.

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Step 1.2.5.1

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

$- 6 \frac{1}{x^{4}}$

Step 1.2.5.2

Combine terms.

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Step 1.2.5.2.1

Combine $- 6$ and $\frac{1}{x^{4}}$ .

$\frac{- 6}{x^{4}}$

Step 1.2.5.2.2

Move the negative in front of the fraction.

$f'' (x) = - \frac{6}{x^{4}}$

Step 1.3

The second derivative of $f (x)$ with respect to $x$ is $- \frac{6}{x^{4}}$ .

$- \frac{6}{x^{4}}$

Step 2

Set the second derivative equal to $0$ then solve the equation $- \frac{6}{x^{4}} = 0$ .

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Step 2.1

Set the second derivative equal to $0$ .

$- \frac{6}{x^{4}} = 0$

Step 2.2

Set the numerator equal to zero.

$6 = 0$

Step 2.3

Since $6 \neq 0$ , there are no solutions.

No solution

Step 3

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

No Inflection Points