Calculus Examples

$f (x) = - \frac{1}{3} x^{6} - \frac{1}{3} x^{3} - \frac{1}{3} x^{- 3} + \frac{1}{2} x^{- 4}$

Step 1

Find the first derivative.

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Step 1.1

By the Sum Rule, the derivative of $- \frac{1}{3} x^{6} - \frac{1}{3} x^{3} - \frac{1}{3} x^{- 3} + \frac{1}{2} x^{- 4}$ with respect to $x$ is $\frac{d}{d x} [- \frac{1}{3} x^{6}] + \frac{d}{d x} [- \frac{1}{3} x^{3}] + \frac{d}{d x} [- \frac{1}{3} x^{- 3}] + \frac{d}{d x} [\frac{1}{2} x^{- 4}]$ .

$\frac{d}{d x} [- \frac{1}{3} x^{6}] + \frac{d}{d x} [- \frac{1}{3} x^{3}] + \frac{d}{d x} [- \frac{1}{3} x^{- 3}] + \frac{d}{d x} [\frac{1}{2} x^{- 4}]$

Step 1.2

Evaluate $\frac{d}{d x} [- \frac{1}{3} x^{6}]$ .

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Step 1.2.1

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

$- \frac{1}{3} \frac{d}{d x} [x^{6}] + \frac{d}{d x} [- \frac{1}{3} x^{3}] + \frac{d}{d x} [- \frac{1}{3} x^{- 3}] + \frac{d}{d x} [\frac{1}{2} x^{- 4}]$

Step 1.2.2

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

$- \frac{1}{3} (6 x^{5}) + \frac{d}{d x} [- \frac{1}{3} x^{3}] + \frac{d}{d x} [- \frac{1}{3} x^{- 3}] + \frac{d}{d x} [\frac{1}{2} x^{- 4}]$

Step 1.2.3

Multiply $6$ by $- 1$ .

$- 6 (\frac{1}{3}) x^{5} + \frac{d}{d x} [- \frac{1}{3} x^{3}] + \frac{d}{d x} [- \frac{1}{3} x^{- 3}] + \frac{d}{d x} [\frac{1}{2} x^{- 4}]$

Step 1.2.4

Combine $- 6$ and $\frac{1}{3}$ .

$\frac{- 6}{3} x^{5} + \frac{d}{d x} [- \frac{1}{3} x^{3}] + \frac{d}{d x} [- \frac{1}{3} x^{- 3}] + \frac{d}{d x} [\frac{1}{2} x^{- 4}]$

Step 1.2.5

Combine $\frac{- 6}{3}$ and $x^{5}$ .

$\frac{- 6 x^{5}}{3} + \frac{d}{d x} [- \frac{1}{3} x^{3}] + \frac{d}{d x} [- \frac{1}{3} x^{- 3}] + \frac{d}{d x} [\frac{1}{2} x^{- 4}]$

Step 1.2.6

Cancel the common factor of $- 6$ and $3$ .

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Step 1.2.6.1

Factor $3$ out of $- 6 x^{5}$ .

$\frac{3 (- 2 x^{5})}{3} + \frac{d}{d x} [- \frac{1}{3} x^{3}] + \frac{d}{d x} [- \frac{1}{3} x^{- 3}] + \frac{d}{d x} [\frac{1}{2} x^{- 4}]$

Step 1.2.6.2

Cancel the common factors.

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Step 1.2.6.2.1

Factor $3$ out of $3$ .

$\frac{3 (- 2 x^{5})}{3 (1)} + \frac{d}{d x} [- \frac{1}{3} x^{3}] + \frac{d}{d x} [- \frac{1}{3} x^{- 3}] + \frac{d}{d x} [\frac{1}{2} x^{- 4}]$

Step 1.2.6.2.2

Cancel the common factor.

$\frac{3 (- 2 x^{5})}{3 \cdot 1} + \frac{d}{d x} [- \frac{1}{3} x^{3}] + \frac{d}{d x} [- \frac{1}{3} x^{- 3}] + \frac{d}{d x} [\frac{1}{2} x^{- 4}]$

Step 1.2.6.2.3

Rewrite the expression.

$\frac{- 2 x^{5}}{1} + \frac{d}{d x} [- \frac{1}{3} x^{3}] + \frac{d}{d x} [- \frac{1}{3} x^{- 3}] + \frac{d}{d x} [\frac{1}{2} x^{- 4}]$

Step 1.2.6.2.4

Divide $- 2 x^{5}$ by $1$ .

$- 2 x^{5} + \frac{d}{d x} [- \frac{1}{3} x^{3}] + \frac{d}{d x} [- \frac{1}{3} x^{- 3}] + \frac{d}{d x} [\frac{1}{2} x^{- 4}]$

Step 1.3

Evaluate $\frac{d}{d x} [- \frac{1}{3} x^{3}]$ .

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Step 1.3.1

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

$- 2 x^{5} - \frac{1}{3} \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- \frac{1}{3} x^{- 3}] + \frac{d}{d x} [\frac{1}{2} x^{- 4}]$

Step 1.3.2

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

$- 2 x^{5} - \frac{1}{3} (3 x^{2}) + \frac{d}{d x} [- \frac{1}{3} x^{- 3}] + \frac{d}{d x} [\frac{1}{2} x^{- 4}]$

Step 1.3.3

Multiply $3$ by $- 1$ .

$- 2 x^{5} - 3 (\frac{1}{3}) x^{2} + \frac{d}{d x} [- \frac{1}{3} x^{- 3}] + \frac{d}{d x} [\frac{1}{2} x^{- 4}]$

Step 1.3.4

Combine $- 3$ and $\frac{1}{3}$ .

$- 2 x^{5} + \frac{- 3}{3} x^{2} + \frac{d}{d x} [- \frac{1}{3} x^{- 3}] + \frac{d}{d x} [\frac{1}{2} x^{- 4}]$

Step 1.3.5

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

$- 2 x^{5} + \frac{- 3 x^{2}}{3} + \frac{d}{d x} [- \frac{1}{3} x^{- 3}] + \frac{d}{d x} [\frac{1}{2} x^{- 4}]$

Step 1.3.6

Cancel the common factor of $- 3$ and $3$ .

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Step 1.3.6.1

Factor $3$ out of $- 3 x^{2}$ .

$- 2 x^{5} + \frac{3 (- x^{2})}{3} + \frac{d}{d x} [- \frac{1}{3} x^{- 3}] + \frac{d}{d x} [\frac{1}{2} x^{- 4}]$

Step 1.3.6.2

Cancel the common factors.

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Step 1.3.6.2.1

Factor $3$ out of $3$ .

$- 2 x^{5} + \frac{3 (- x^{2})}{3 (1)} + \frac{d}{d x} [- \frac{1}{3} x^{- 3}] + \frac{d}{d x} [\frac{1}{2} x^{- 4}]$

Step 1.3.6.2.2

Cancel the common factor.

$- 2 x^{5} + \frac{3 (- x^{2})}{3 \cdot 1} + \frac{d}{d x} [- \frac{1}{3} x^{- 3}] + \frac{d}{d x} [\frac{1}{2} x^{- 4}]$

Step 1.3.6.2.3

Rewrite the expression.

$- 2 x^{5} + \frac{- x^{2}}{1} + \frac{d}{d x} [- \frac{1}{3} x^{- 3}] + \frac{d}{d x} [\frac{1}{2} x^{- 4}]$

Step 1.3.6.2.4

Divide $- x^{2}$ by $1$ .

$- 2 x^{5} - x^{2} + \frac{d}{d x} [- \frac{1}{3} x^{- 3}] + \frac{d}{d x} [\frac{1}{2} x^{- 4}]$

Step 1.4

Evaluate $\frac{d}{d x} [- \frac{1}{3} x^{- 3}]$ .

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Step 1.4.1

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

$- 2 x^{5} - x^{2} - \frac{1}{3} \frac{d}{d x} [x^{- 3}] + \frac{d}{d x} [\frac{1}{2} x^{- 4}]$

Step 1.4.2

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

$- 2 x^{5} - x^{2} - \frac{1}{3} (- 3 x^{- 4}) + \frac{d}{d x} [\frac{1}{2} x^{- 4}]$

Step 1.4.3

Multiply $- 3$ by $- 1$ .

$- 2 x^{5} - x^{2} + 3 (\frac{1}{3}) x^{- 4} + \frac{d}{d x} [\frac{1}{2} x^{- 4}]$

Step 1.4.4

Combine $3$ and $\frac{1}{3}$ .

$- 2 x^{5} - x^{2} + \frac{3}{3} x^{- 4} + \frac{d}{d x} [\frac{1}{2} x^{- 4}]$

Step 1.4.5

Combine $\frac{3}{3}$ and $x^{- 4}$ .

$- 2 x^{5} - x^{2} + \frac{3 x^{- 4}}{3} + \frac{d}{d x} [\frac{1}{2} x^{- 4}]$

Step 1.4.6

Move $x^{- 4}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- 2 x^{5} - x^{2} + \frac{3}{3 x^{4}} + \frac{d}{d x} [\frac{1}{2} x^{- 4}]$

Step 1.4.7

Cancel the common factor of $3$ .

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Step 1.4.7.1

Cancel the common factor.

$- 2 x^{5} - x^{2} + \frac{3}{3 x^{4}} + \frac{d}{d x} [\frac{1}{2} x^{- 4}]$

Step 1.4.7.2

Rewrite the expression.

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

Step 1.5

Evaluate $\frac{d}{d x} [\frac{1}{2} x^{- 4}]$ .

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Step 1.5.1

Since $\frac{1}{2}$ is constant with respect to $x$ , the derivative of $\frac{1}{2} x^{- 4}$ with respect to $x$ is $\frac{1}{2} \frac{d}{d x} [x^{- 4}]$ .

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

Step 1.5.2

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

$- 2 x^{5} - x^{2} + \frac{1}{x^{4}} + \frac{1}{2} (- 4 x^{- 5})$

Step 1.5.3

Combine $- 4$ and $\frac{1}{2}$ .

$- 2 x^{5} - x^{2} + \frac{1}{x^{4}} + \frac{- 4}{2} x^{- 5}$

Step 1.5.4

Combine $\frac{- 4}{2}$ and $x^{- 5}$ .

$- 2 x^{5} - x^{2} + \frac{1}{x^{4}} + \frac{- 4 x^{- 5}}{2}$

Step 1.5.5

Move $x^{- 5}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- 2 x^{5} - x^{2} + \frac{1}{x^{4}} + \frac{- 4}{2 x^{5}}$

Step 1.5.6

Cancel the common factor of $- 4$ and $2$ .

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Step 1.5.6.1

Factor $2$ out of $- 4$ .

$- 2 x^{5} - x^{2} + \frac{1}{x^{4}} + \frac{2 \cdot - 2}{2 x^{5}}$

Step 1.5.6.2

Cancel the common factors.

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Step 1.5.6.2.1

Factor $2$ out of $2 x^{5}$ .

$- 2 x^{5} - x^{2} + \frac{1}{x^{4}} + \frac{2 \cdot - 2}{2 (x^{5})}$

Step 1.5.6.2.2

Cancel the common factor.

$- 2 x^{5} - x^{2} + \frac{1}{x^{4}} + \frac{2 \cdot - 2}{2 x^{5}}$

Step 1.5.6.2.3

Rewrite the expression.

$- 2 x^{5} - x^{2} + \frac{1}{x^{4}} + \frac{- 2}{x^{5}}$

Step 1.5.7

Move the negative in front of the fraction.

$f' (x) = - 2 x^{5} - x^{2} + \frac{1}{x^{4}} - \frac{2}{x^{5}}$

Step 2

Find the second derivative.

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Step 2.1

By the Sum Rule, the derivative of $- 2 x^{5} - x^{2} + \frac{1}{x^{4}} - \frac{2}{x^{5}}$ with respect to $x$ is $\frac{d}{d x} [- 2 x^{5}] + \frac{d}{d x} [- x^{2}] + \frac{d}{d x} [\frac{1}{x^{4}}] + \frac{d}{d x} [- \frac{2}{x^{5}}]$ .

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

Step 2.2

Evaluate $\frac{d}{d x} [- 2 x^{5}]$ .

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Step 2.2.1

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2 x^{5}$ with respect to $x$ is $- 2 \frac{d}{d x} [x^{5}]$ .

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

Step 2.2.2

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

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

Step 2.2.3

Multiply $5$ by $- 2$ .

$- 10 x^{4} + \frac{d}{d x} [- x^{2}] + \frac{d}{d x} [\frac{1}{x^{4}}] + \frac{d}{d x} [- \frac{2}{x^{5}}]$

Step 2.3

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

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Step 2.3.1

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

$- 10 x^{4} - \frac{d}{d x} [x^{2}] + \frac{d}{d x} [\frac{1}{x^{4}}] + \frac{d}{d x} [- \frac{2}{x^{5}}]$

Step 2.3.2

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

$- 10 x^{4} - (2 x) + \frac{d}{d x} [\frac{1}{x^{4}}] + \frac{d}{d x} [- \frac{2}{x^{5}}]$

Step 2.3.3

Multiply $2$ by $- 1$ .

$- 10 x^{4} - 2 x + \frac{d}{d x} [\frac{1}{x^{4}}] + \frac{d}{d x} [- \frac{2}{x^{5}}]$

Step 2.4

Evaluate $\frac{d}{d x} [\frac{1}{x^{4}}]$ .

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Step 2.4.1

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

$- 10 x^{4} - 2 x + \frac{d}{d x} [{(x^{4})}^{- 1}] + \frac{d}{d x} [- \frac{2}{x^{5}}]$

Step 2.4.2

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 $g (x) = x^{4}$ .

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Step 2.4.2.1

To apply the Chain Rule, set $u_{1}$ as $x^{4}$ .

$- 10 x^{4} - 2 x + \frac{d}{d u_{1}} [{u_{1}}^{- 1}] \frac{d}{d x} [x^{4}] + \frac{d}{d x} [- \frac{2}{x^{5}}]$

Step 2.4.2.2

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

$- 10 x^{4} - 2 x - {u_{1}}^{- 2} \frac{d}{d x} [x^{4}] + \frac{d}{d x} [- \frac{2}{x^{5}}]$

Step 2.4.2.3

Replace all occurrences of $u_{1}$ with $x^{4}$ .

$- 10 x^{4} - 2 x - {(x^{4})}^{- 2} \frac{d}{d x} [x^{4}] + \frac{d}{d x} [- \frac{2}{x^{5}}]$

Step 2.4.3

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

$- 10 x^{4} - 2 x - {(x^{4})}^{- 2} (4 x^{3}) + \frac{d}{d x} [- \frac{2}{x^{5}}]$

Step 2.4.4

Multiply the exponents in ${(x^{4})}^{- 2}$ .

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Step 2.4.4.1

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

$- 10 x^{4} - 2 x - x^{4 \cdot - 2} (4 x^{3}) + \frac{d}{d x} [- \frac{2}{x^{5}}]$

Step 2.4.4.2

Multiply $4$ by $- 2$ .

$- 10 x^{4} - 2 x - x^{- 8} (4 x^{3}) + \frac{d}{d x} [- \frac{2}{x^{5}}]$

Step 2.4.5

Multiply $4$ by $- 1$ .

$- 10 x^{4} - 2 x - 4 x^{- 8} x^{3} + \frac{d}{d x} [- \frac{2}{x^{5}}]$

Step 2.4.6

Multiply $x^{- 8}$ by $x^{3}$ by adding the exponents.

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Step 2.4.6.1

Move $x^{3}$ .

$- 10 x^{4} - 2 x - 4 (x^{3} x^{- 8}) + \frac{d}{d x} [- \frac{2}{x^{5}}]$

Step 2.4.6.2

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

$- 10 x^{4} - 2 x - 4 x^{3 - 8} + \frac{d}{d x} [- \frac{2}{x^{5}}]$

Step 2.4.6.3

Subtract $8$ from $3$ .

$- 10 x^{4} - 2 x - 4 x^{- 5} + \frac{d}{d x} [- \frac{2}{x^{5}}]$

Step 2.5

Evaluate $\frac{d}{d x} [- \frac{2}{x^{5}}]$ .

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Step 2.5.1

Since $- 2$ is constant with respect to $x$ , the derivative of $- \frac{2}{x^{5}}$ with respect to $x$ is $- 2 \frac{d}{d x} [\frac{1}{x^{5}}]$ .

$- 10 x^{4} - 2 x - 4 x^{- 5} - 2 \frac{d}{d x} [\frac{1}{x^{5}}]$

Step 2.5.2

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

$- 10 x^{4} - 2 x - 4 x^{- 5} - 2 \frac{d}{d x} [{(x^{5})}^{- 1}]$

Step 2.5.3

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 $g (x) = x^{5}$ .

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Step 2.5.3.1

To apply the Chain Rule, set $u_{2}$ as $x^{5}$ .

$- 10 x^{4} - 2 x - 4 x^{- 5} - 2 (\frac{d}{d u_{2}} [{u_{2}}^{- 1}] \frac{d}{d x} [x^{5}])$

Step 2.5.3.2

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

$- 10 x^{4} - 2 x - 4 x^{- 5} - 2 (- {u_{2}}^{- 2} \frac{d}{d x} [x^{5}])$

Step 2.5.3.3

Replace all occurrences of $u_{2}$ with $x^{5}$ .

$- 10 x^{4} - 2 x - 4 x^{- 5} - 2 (- {(x^{5})}^{- 2} \frac{d}{d x} [x^{5}])$

Step 2.5.4

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

$- 10 x^{4} - 2 x - 4 x^{- 5} - 2 (- {(x^{5})}^{- 2} (5 x^{4}))$

Step 2.5.5

Multiply the exponents in ${(x^{5})}^{- 2}$ .

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Step 2.5.5.1

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

$- 10 x^{4} - 2 x - 4 x^{- 5} - 2 (- x^{5 \cdot - 2} (5 x^{4}))$

Step 2.5.5.2

Multiply $5$ by $- 2$ .

$- 10 x^{4} - 2 x - 4 x^{- 5} - 2 (- x^{- 10} (5 x^{4}))$

Step 2.5.6

Multiply $5$ by $- 1$ .

$- 10 x^{4} - 2 x - 4 x^{- 5} - 2 (- 5 x^{- 10} x^{4})$

Step 2.5.7

Multiply $x^{- 10}$ by $x^{4}$ by adding the exponents.

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Step 2.5.7.1

Move $x^{4}$ .

$- 10 x^{4} - 2 x - 4 x^{- 5} - 2 (- 5 (x^{4} x^{- 10}))$

Step 2.5.7.2

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

$- 10 x^{4} - 2 x - 4 x^{- 5} - 2 (- 5 x^{4 - 10})$

Step 2.5.7.3

Subtract $10$ from $4$ .

$- 10 x^{4} - 2 x - 4 x^{- 5} - 2 (- 5 x^{- 6})$

Step 2.5.8

Multiply $- 5$ by $- 2$ .

$- 10 x^{4} - 2 x - 4 x^{- 5} + 10 x^{- 6}$

Step 2.6

Simplify.

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Step 2.6.1

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

$- 10 x^{4} - 2 x - 4 \frac{1}{x^{5}} + 10 x^{- 6}$

Step 2.6.2

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

$- 10 x^{4} - 2 x - 4 \frac{1}{x^{5}} + 10 \frac{1}{x^{6}}$

Step 2.6.3

Combine terms.

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Step 2.6.3.1

Combine $- 4$ and $\frac{1}{x^{5}}$ .

$- 10 x^{4} - 2 x + \frac{- 4}{x^{5}} + 10 \frac{1}{x^{6}}$

Step 2.6.3.2

Move the negative in front of the fraction.

$- 10 x^{4} - 2 x - \frac{4}{x^{5}} + 10 \frac{1}{x^{6}}$

Step 2.6.3.3

Combine $10$ and $\frac{1}{x^{6}}$ .

$f'' (x) = - 10 x^{4} - 2 x - \frac{4}{x^{5}} + \frac{10}{x^{6}}$

Step 3

The second derivative of $f (x)$ with respect to $x$ is $- 10 x^{4} - 2 x - \frac{4}{x^{5}} + \frac{10}{x^{6}}$ .

$- 10 x^{4} - 2 x - \frac{4}{x^{5}} + \frac{10}{x^{6}}$