Calculus Examples

$y = \frac{1}{x^{10}}$

Step 1

Find the first derivative.

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Step 1.1

Apply basic rules of exponents.

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Step 1.1.1

Rewrite $\frac{1}{x^{10}}$ as ${(x^{10})}^{- 1}$ .

$\frac{d}{d x} [{(x^{10})}^{- 1}]$

Step 1.1.2

Multiply the exponents in ${(x^{10})}^{- 1}$ .

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Step 1.1.2.1

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

$\frac{d}{d x} [x^{10 \cdot - 1}]$

Step 1.1.2.2

Multiply $10$ by $- 1$ .

$\frac{d}{d x} [x^{- 10}]$

Step 1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - 10$ .

$- 10 x^{- 11}$

Step 1.3

Simplify.

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Step 1.3.1

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

$- 10 \frac{1}{x^{11}}$

Step 1.3.2

Combine terms.

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Step 1.3.2.1

Combine $- 10$ and $\frac{1}{x^{11}}$ .

$\frac{- 10}{x^{11}}$

Step 1.3.2.2

Move the negative in front of the fraction.

$f' (x) = - \frac{10}{x^{11}}$

Step 2

Find the second derivative.

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Step 2.1

Since $- 10$ is constant with respect to $x$ , the derivative of $- \frac{10}{x^{11}}$ with respect to $x$ is $- 10 \frac{d}{d x} [\frac{1}{x^{11}}]$ .

$- 10 \frac{d}{d x} [\frac{1}{x^{11}}]$

Step 2.2

Apply basic rules of exponents.

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Step 2.2.1

Rewrite $\frac{1}{x^{11}}$ as ${(x^{11})}^{- 1}$ .

$- 10 \frac{d}{d x} [{(x^{11})}^{- 1}]$

Step 2.2.2

Multiply the exponents in ${(x^{11})}^{- 1}$ .

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Step 2.2.2.1

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

$- 10 \frac{d}{d x} [x^{11 \cdot - 1}]$

Step 2.2.2.2

Multiply $11$ by $- 1$ .

$- 10 \frac{d}{d x} [x^{- 11}]$

Step 2.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - 11$ .

$- 10 (- 11 x^{- 12})$

Step 2.4

Multiply $- 11$ by $- 10$ .

$110 x^{- 12}$

Step 2.5

Simplify.

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Step 2.5.1

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

$110 \frac{1}{x^{12}}$

Step 2.5.2

Combine $110$ and $\frac{1}{x^{12}}$ .

$f'' (x) = \frac{110}{x^{12}}$

Step 3

Find the third derivative.

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Step 3.1

Since $110$ is constant with respect to $x$ , the derivative of $\frac{110}{x^{12}}$ with respect to $x$ is $110 \frac{d}{d x} [\frac{1}{x^{12}}]$ .

$110 \frac{d}{d x} [\frac{1}{x^{12}}]$

Step 3.2

Apply basic rules of exponents.

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Step 3.2.1

Rewrite $\frac{1}{x^{12}}$ as ${(x^{12})}^{- 1}$ .

$110 \frac{d}{d x} [{(x^{12})}^{- 1}]$

Step 3.2.2

Multiply the exponents in ${(x^{12})}^{- 1}$ .

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Step 3.2.2.1

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

$110 \frac{d}{d x} [x^{12 \cdot - 1}]$

Step 3.2.2.2

Multiply $12$ by $- 1$ .

$110 \frac{d}{d x} [x^{- 12}]$

Step 3.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - 12$ .

$110 (- 12 x^{- 13})$

Step 3.4

Multiply $- 12$ by $110$ .

$- 1320 x^{- 13}$

Step 3.5

Simplify.

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Step 3.5.1

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

$- 1320 \frac{1}{x^{13}}$

Step 3.5.2

Combine terms.

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Step 3.5.2.1

Combine $- 1320$ and $\frac{1}{x^{13}}$ .

$\frac{- 1320}{x^{13}}$

Step 3.5.2.2

Move the negative in front of the fraction.

$f''' (x) = - \frac{1320}{x^{13}}$

Step 4

Find the fourth derivative.

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Step 4.1

Since $- 1320$ is constant with respect to $x$ , the derivative of $- \frac{1320}{x^{13}}$ with respect to $x$ is $- 1320 \frac{d}{d x} [\frac{1}{x^{13}}]$ .

$- 1320 \frac{d}{d x} [\frac{1}{x^{13}}]$

Step 4.2

Apply basic rules of exponents.

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Step 4.2.1

Rewrite $\frac{1}{x^{13}}$ as ${(x^{13})}^{- 1}$ .

$- 1320 \frac{d}{d x} [{(x^{13})}^{- 1}]$

Step 4.2.2

Multiply the exponents in ${(x^{13})}^{- 1}$ .

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Step 4.2.2.1

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

$- 1320 \frac{d}{d x} [x^{13 \cdot - 1}]$

Step 4.2.2.2

Multiply $13$ by $- 1$ .

$- 1320 \frac{d}{d x} [x^{- 13}]$

Step 4.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - 13$ .

$- 1320 (- 13 x^{- 14})$

Step 4.4

Multiply $- 13$ by $- 1320$ .

$17160 x^{- 14}$

Step 4.5

Simplify.

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Step 4.5.1

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

$17160 \frac{1}{x^{14}}$

Step 4.5.2

Combine $17160$ and $\frac{1}{x^{14}}$ .

$f^{4} (x) = \frac{17160}{x^{14}}$