Calculus Examples

$y = x^{- 7} + x^{7}$

Step 1

Find the first derivative.

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Step 1.1

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

$\frac{d}{d x} [x^{- 7}] + \frac{d}{d x} [x^{7}]$

Step 1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - 7$ .

$- 7 x^{- 8} + \frac{d}{d x} [x^{7}]$

Step 1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 7$ .

$- 7 x^{- 8} + 7 x^{6}$

Step 1.4

Simplify.

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Step 1.4.1

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

$- 7 \frac{1}{x^{8}} + 7 x^{6}$

Step 1.4.2

Combine terms.

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Step 1.4.2.1

Combine $- 7$ and $\frac{1}{x^{8}}$ .

$\frac{- 7}{x^{8}} + 7 x^{6}$

Step 1.4.2.2

Move the negative in front of the fraction.

$- \frac{7}{x^{8}} + 7 x^{6}$

Step 1.4.3

Reorder terms.

$f' (x) = 7 x^{6} - \frac{7}{x^{8}}$

Step 2

Find the second derivative.

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Step 2.1

By the Sum Rule, the derivative of $7 x^{6} - \frac{7}{x^{8}}$ with respect to $x$ is $\frac{d}{d x} [7 x^{6}] + \frac{d}{d x} [- \frac{7}{x^{8}}]$ .

$\frac{d}{d x} [7 x^{6}] + \frac{d}{d x} [- \frac{7}{x^{8}}]$

Step 2.2

Evaluate $\frac{d}{d x} [7 x^{6}]$ .

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Step 2.2.1

Since $7$ is constant with respect to $x$ , the derivative of $7 x^{6}$ with respect to $x$ is $7 \frac{d}{d x} [x^{6}]$ .

$7 \frac{d}{d x} [x^{6}] + \frac{d}{d x} [- \frac{7}{x^{8}}]$

Step 2.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 6$ .

$7 (6 x^{5}) + \frac{d}{d x} [- \frac{7}{x^{8}}]$

Step 2.2.3

Multiply $6$ by $7$ .

$42 x^{5} + \frac{d}{d x} [- \frac{7}{x^{8}}]$

Step 2.3

Evaluate $\frac{d}{d x} [- \frac{7}{x^{8}}]$ .

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Step 2.3.1

Since $- 7$ is constant with respect to $x$ , the derivative of $- \frac{7}{x^{8}}$ with respect to $x$ is $- 7 \frac{d}{d x} [\frac{1}{x^{8}}]$ .

$42 x^{5} - 7 \frac{d}{d x} [\frac{1}{x^{8}}]$

Step 2.3.2

Rewrite $\frac{1}{x^{8}}$ as ${(x^{8})}^{- 1}$ .

$42 x^{5} - 7 \frac{d}{d x} [{(x^{8})}^{- 1}]$

Step 2.3.3

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 $g (x) = x^{8}$ .

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Step 2.3.3.1

To apply the Chain Rule, set $u$ as $x^{8}$ .

$42 x^{5} - 7 (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{8}])$

Step 2.3.3.2

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

$42 x^{5} - 7 (- u^{- 2} \frac{d}{d x} [x^{8}])$

Step 2.3.3.3

Replace all occurrences of $u$ with $x^{8}$ .

$42 x^{5} - 7 (- {(x^{8})}^{- 2} \frac{d}{d x} [x^{8}])$

Step 2.3.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 8$ .

$42 x^{5} - 7 (- {(x^{8})}^{- 2} (8 x^{7}))$

Step 2.3.5

Multiply the exponents in ${(x^{8})}^{- 2}$ .

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Step 2.3.5.1

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

$42 x^{5} - 7 (- x^{8 \cdot - 2} (8 x^{7}))$

Step 2.3.5.2

Multiply $8$ by $- 2$ .

$42 x^{5} - 7 (- x^{- 16} (8 x^{7}))$

Step 2.3.6

Multiply $8$ by $- 1$ .

$42 x^{5} - 7 (- 8 x^{- 16} x^{7})$

Step 2.3.7

Multiply $x^{- 16}$ by $x^{7}$ by adding the exponents.

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Step 2.3.7.1

Move $x^{7}$ .

$42 x^{5} - 7 (- 8 (x^{7} x^{- 16}))$

Step 2.3.7.2

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

$42 x^{5} - 7 (- 8 x^{7 - 16})$

Step 2.3.7.3

Subtract $16$ from $7$ .

$42 x^{5} - 7 (- 8 x^{- 9})$

Step 2.3.8

Multiply $- 8$ by $- 7$ .

$42 x^{5} + 56 x^{- 9}$

Step 2.4

Simplify.

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Step 2.4.1

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

$42 x^{5} + 56 \frac{1}{x^{9}}$

Step 2.4.2

Combine $56$ and $\frac{1}{x^{9}}$ .

$f'' (x) = 42 x^{5} + \frac{56}{x^{9}}$

Step 3

Find the third derivative.

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Step 3.1

By the Sum Rule, the derivative of $42 x^{5} + \frac{56}{x^{9}}$ with respect to $x$ is $\frac{d}{d x} [42 x^{5}] + \frac{d}{d x} [\frac{56}{x^{9}}]$ .

$\frac{d}{d x} [42 x^{5}] + \frac{d}{d x} [\frac{56}{x^{9}}]$

Step 3.2

Evaluate $\frac{d}{d x} [42 x^{5}]$ .

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Step 3.2.1

Since $42$ is constant with respect to $x$ , the derivative of $42 x^{5}$ with respect to $x$ is $42 \frac{d}{d x} [x^{5}]$ .

$42 \frac{d}{d x} [x^{5}] + \frac{d}{d x} [\frac{56}{x^{9}}]$

Step 3.2.2

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

$42 (5 x^{4}) + \frac{d}{d x} [\frac{56}{x^{9}}]$

Step 3.2.3

Multiply $5$ by $42$ .

$210 x^{4} + \frac{d}{d x} [\frac{56}{x^{9}}]$

Step 3.3

Evaluate $\frac{d}{d x} [\frac{56}{x^{9}}]$ .

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Step 3.3.1

Since $56$ is constant with respect to $x$ , the derivative of $\frac{56}{x^{9}}$ with respect to $x$ is $56 \frac{d}{d x} [\frac{1}{x^{9}}]$ .

$210 x^{4} + 56 \frac{d}{d x} [\frac{1}{x^{9}}]$

Step 3.3.2

Rewrite $\frac{1}{x^{9}}$ as ${(x^{9})}^{- 1}$ .

$210 x^{4} + 56 \frac{d}{d x} [{(x^{9})}^{- 1}]$

Step 3.3.3

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 $g (x) = x^{9}$ .

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Step 3.3.3.1

To apply the Chain Rule, set $u$ as $x^{9}$ .

$210 x^{4} + 56 (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{9}])$

Step 3.3.3.2

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

$210 x^{4} + 56 (- u^{- 2} \frac{d}{d x} [x^{9}])$

Step 3.3.3.3

Replace all occurrences of $u$ with $x^{9}$ .

$210 x^{4} + 56 (- {(x^{9})}^{- 2} \frac{d}{d x} [x^{9}])$

Step 3.3.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 9$ .

$210 x^{4} + 56 (- {(x^{9})}^{- 2} (9 x^{8}))$

Step 3.3.5

Multiply the exponents in ${(x^{9})}^{- 2}$ .

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Step 3.3.5.1

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

$210 x^{4} + 56 (- x^{9 \cdot - 2} (9 x^{8}))$

Step 3.3.5.2

Multiply $9$ by $- 2$ .

$210 x^{4} + 56 (- x^{- 18} (9 x^{8}))$

Step 3.3.6

Multiply $9$ by $- 1$ .

$210 x^{4} + 56 (- 9 x^{- 18} x^{8})$

Step 3.3.7

Multiply $x^{- 18}$ by $x^{8}$ by adding the exponents.

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Step 3.3.7.1

Move $x^{8}$ .

$210 x^{4} + 56 (- 9 (x^{8} x^{- 18}))$

Step 3.3.7.2

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

$210 x^{4} + 56 (- 9 x^{8 - 18})$

Step 3.3.7.3

Subtract $18$ from $8$ .

$210 x^{4} + 56 (- 9 x^{- 10})$

Step 3.3.8

Multiply $- 9$ by $56$ .

$210 x^{4} - 504 x^{- 10}$

Step 3.4

Simplify.

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Step 3.4.1

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

$210 x^{4} - 504 \frac{1}{x^{10}}$

Step 3.4.2

Combine terms.

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Step 3.4.2.1

Combine $- 504$ and $\frac{1}{x^{10}}$ .

$210 x^{4} + \frac{- 504}{x^{10}}$

Step 3.4.2.2

Move the negative in front of the fraction.

$f''' (x) = 210 x^{4} - \frac{504}{x^{10}}$

Step 4

Find the fourth derivative.

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Step 4.1

By the Sum Rule, the derivative of $210 x^{4} - \frac{504}{x^{10}}$ with respect to $x$ is $\frac{d}{d x} [210 x^{4}] + \frac{d}{d x} [- \frac{504}{x^{10}}]$ .

$\frac{d}{d x} [210 x^{4}] + \frac{d}{d x} [- \frac{504}{x^{10}}]$

Step 4.2

Evaluate $\frac{d}{d x} [210 x^{4}]$ .

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Step 4.2.1

Since $210$ is constant with respect to $x$ , the derivative of $210 x^{4}$ with respect to $x$ is $210 \frac{d}{d x} [x^{4}]$ .

$210 \frac{d}{d x} [x^{4}] + \frac{d}{d x} [- \frac{504}{x^{10}}]$

Step 4.2.2

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

$210 (4 x^{3}) + \frac{d}{d x} [- \frac{504}{x^{10}}]$

Step 4.2.3

Multiply $4$ by $210$ .

$840 x^{3} + \frac{d}{d x} [- \frac{504}{x^{10}}]$

Step 4.3

Evaluate $\frac{d}{d x} [- \frac{504}{x^{10}}]$ .

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Step 4.3.1

Since $- 504$ is constant with respect to $x$ , the derivative of $- \frac{504}{x^{10}}$ with respect to $x$ is $- 504 \frac{d}{d x} [\frac{1}{x^{10}}]$ .

$840 x^{3} - 504 \frac{d}{d x} [\frac{1}{x^{10}}]$

Step 4.3.2

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

$840 x^{3} - 504 \frac{d}{d x} [{(x^{10})}^{- 1}]$

Step 4.3.3

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 $g (x) = x^{10}$ .

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Step 4.3.3.1

To apply the Chain Rule, set $u$ as $x^{10}$ .

$840 x^{3} - 504 (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{10}])$

Step 4.3.3.2

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

$840 x^{3} - 504 (- u^{- 2} \frac{d}{d x} [x^{10}])$

Step 4.3.3.3

Replace all occurrences of $u$ with $x^{10}$ .

$840 x^{3} - 504 (- {(x^{10})}^{- 2} \frac{d}{d x} [x^{10}])$

Step 4.3.4

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

$840 x^{3} - 504 (- {(x^{10})}^{- 2} (10 x^{9}))$

Step 4.3.5

Multiply the exponents in ${(x^{10})}^{- 2}$ .

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Step 4.3.5.1

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

$840 x^{3} - 504 (- x^{10 \cdot - 2} (10 x^{9}))$

Step 4.3.5.2

Multiply $10$ by $- 2$ .

$840 x^{3} - 504 (- x^{- 20} (10 x^{9}))$

Step 4.3.6

Multiply $10$ by $- 1$ .

$840 x^{3} - 504 (- 10 x^{- 20} x^{9})$

Step 4.3.7

Multiply $x^{- 20}$ by $x^{9}$ by adding the exponents.

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Step 4.3.7.1

Move $x^{9}$ .

$840 x^{3} - 504 (- 10 (x^{9} x^{- 20}))$

Step 4.3.7.2

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

$840 x^{3} - 504 (- 10 x^{9 - 20})$

Step 4.3.7.3

Subtract $20$ from $9$ .

$840 x^{3} - 504 (- 10 x^{- 11})$

Step 4.3.8

Multiply $- 10$ by $- 504$ .

$840 x^{3} + 5040 x^{- 11}$

Step 4.4

Simplify.

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Step 4.4.1

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

$840 x^{3} + 5040 \frac{1}{x^{11}}$

Step 4.4.2

Combine $5040$ and $\frac{1}{x^{11}}$ .

$f^{4} (x) = 840 x^{3} + \frac{5040}{x^{11}}$