Calculus Examples

$y = - \frac{3}{x^{2}}$

Step 1

Find the first derivative.

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

$f' (x) = - 3 \frac{d}{d x} \frac{1}{x^{2}}$

Apply basic rules of exponents.

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

$f' (x) = - 3 \frac{d}{d x} {(x^{2})}^{- 1}$

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

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

$f' (x) = - 3 \frac{d}{d x} x^{2 \cdot - 1}$

Multiply $2$ by $- 1$ .

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

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

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

Multiply $- 2$ by $- 3$ .

$f' (x) = 6 x^{- 3}$

Simplify.

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

$f' (x) = 6 (\frac{1}{x^{3}})$

Combine $6$ and $\frac{1}{x^{3}}$ .

$f' (x) = \frac{6}{x^{3}}$

Step 2

Find the second derivative.

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Since $6$ is constant with respect to $x$ , the derivative of $\frac{6}{x^{3}}$ with respect to $x$ is $6 \frac{d}{d x} [\frac{1}{x^{3}}]$ .

$f'' (x) = 6 \frac{d}{d x} (\frac{1}{x^{3}})$

Apply basic rules of exponents.

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

$f'' (x) = 6 \frac{d}{d x} ({(x^{3})}^{- 1})$

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

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

$f'' (x) = 6 \frac{d}{d x} (x^{3 \cdot - 1})$

Multiply $3$ by $- 1$ .

$f'' (x) = 6 \frac{d}{d x} (x^{- 3})$

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

$f'' (x) = 6 (- 3 x^{- 4})$

Multiply $- 3$ by $6$ .

$f'' (x) = - 18 x^{- 4}$

Simplify.

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

$f'' (x) = - 18 \frac{1}{x^{4}}$

Combine terms.

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Combine $- 18$ and $\frac{1}{x^{4}}$ .

$f'' (x) = \frac{- 18}{x^{4}}$

Move the negative in front of the fraction.

$f'' (x) = - \frac{18}{x^{4}}$

Step 3

Find the third derivative.

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Since $- 18$ is constant with respect to $x$ , the derivative of $- \frac{18}{x^{4}}$ with respect to $x$ is $- 18 \frac{d}{d x} [\frac{1}{x^{4}}]$ .

$f''' (x) = - 18 \frac{d}{d x} \frac{1}{x^{4}}$

Apply basic rules of exponents.

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Rewrite $\frac{1}{x^{4}}$ as ${(x^{4})}^{- 1}$ .

$f''' (x) = - 18 \frac{d}{d x} {(x^{4})}^{- 1}$

Multiply the exponents in ${(x^{4})}^{- 1}$ .

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

$f''' (x) = - 18 \frac{d}{d x} x^{4 \cdot - 1}$

Multiply $4$ by $- 1$ .

$f''' (x) = - 18 \frac{d}{d x} x^{- 4}$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - 4$ .

$f''' (x) = - 18 (- 4 x^{- 5})$

Multiply $- 4$ by $- 18$ .

$f''' (x) = 72 x^{- 5}$

Simplify.

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

$f''' (x) = 72 (\frac{1}{x^{5}})$

Combine $72$ and $\frac{1}{x^{5}}$ .

$f''' (x) = \frac{72}{x^{5}}$

Step 4

Find the fourth derivative.

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Since $72$ is constant with respect to $x$ , the derivative of $\frac{72}{x^{5}}$ with respect to $x$ is $72 \frac{d}{d x} [\frac{1}{x^{5}}]$ .

$f^{4} (x) = 72 \frac{d}{d x} (\frac{1}{x^{5}})$

Apply basic rules of exponents.

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Rewrite $\frac{1}{x^{5}}$ as ${(x^{5})}^{- 1}$ .

$f^{4} (x) = 72 \frac{d}{d x} ({(x^{5})}^{- 1})$

Multiply the exponents in ${(x^{5})}^{- 1}$ .

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

$f^{4} (x) = 72 \frac{d}{d x} (x^{5 \cdot - 1})$

Multiply $5$ by $- 1$ .

$f^{4} (x) = 72 \frac{d}{d x} (x^{- 5})$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - 5$ .

$f^{4} (x) = 72 (- 5 x^{- 6})$

Multiply $- 5$ by $72$ .

$f^{4} (x) = - 360 x^{- 6}$

Simplify.

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

$f^{4} (x) = - 360 \frac{1}{x^{6}}$

Combine terms.

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Combine $- 360$ and $\frac{1}{x^{6}}$ .

$f^{4} (x) = \frac{- 360}{x^{6}}$

Move the negative in front of the fraction.

$f^{4} (x) = - \frac{360}{x^{6}}$