Calculus Examples

$f (x) = \arctan (x^{5})$

Step 1

Find the first derivative of the function.

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Step 1.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \arctan (x)$ and $g (x) = x^{5}$ .

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Step 1.1.1

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

$\frac{d}{d u} [\arctan (u)] \frac{d}{d x} [x^{5}]$

Step 1.1.2

The derivative of $\arctan (u)$ with respect to $u$ is $\frac{1}{1 + u^{2}}$ .

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

Step 1.1.3

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

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

Step 1.2

Differentiate using the Power Rule.

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Step 1.2.1

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

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Step 1.2.1.1

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

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

Step 1.2.1.2

Multiply $5$ by $2$ .

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

Step 1.2.2

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

$\frac{1}{1 + x^{10}} (5 x^{4})$

Step 1.2.3

Combine fractions.

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Step 1.2.3.1

Combine $5$ and $\frac{1}{1 + x^{10}}$ .

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

Step 1.2.3.2

Combine $\frac{5}{1 + x^{10}}$ and $x^{4}$ .

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

Step 1.2.3.3

Reorder terms.

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

Step 2

Find the second derivative of the function.

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Step 2.1

Since $5$ is constant with respect to $x$ , the derivative of $\frac{5 x^{4}}{x^{10} + 1}$ with respect to $x$ is $5 \frac{d}{d x} [\frac{x^{4}}{x^{10} + 1}]$ .

$f'' (x) = 5 \frac{d}{d x} (\frac{x^{4}}{x^{10} + 1})$

Step 2.2

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = x^{4}$ and $g (x) = x^{10} + 1$ .

$f'' (x) = 5 (\frac{(x^{10} + 1) \frac{d}{d x} (x^{4}) - x^{4} \frac{d}{d x} (x^{10} + 1)}{{(x^{10} + 1)}^{2}})$

Step 2.3

Differentiate.

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Step 2.3.1

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

$f'' (x) = 5 (\frac{(x^{10} + 1) (4 x^{3}) - x^{4} \frac{d}{d x} (x^{10} + 1)}{{(x^{10} + 1)}^{2}})$

Step 2.3.2

Move $4$ to the left of $x^{10} + 1$ .

$f'' (x) = 5 (\frac{4 \cdot ((x^{10} + 1) x^{3}) - x^{4} \frac{d}{d x} (x^{10} + 1)}{{(x^{10} + 1)}^{2}})$

Step 2.3.3

By the Sum Rule, the derivative of $x^{10} + 1$ with respect to $x$ is $\frac{d}{d x} [x^{10}] + \frac{d}{d x} [1]$ .

$f'' (x) = 5 (\frac{4 (x^{10} + 1) x^{3} - x^{4} (\frac{d}{d x} (x^{10}) + \frac{d}{d x} (1))}{{(x^{10} + 1)}^{2}})$

Step 2.3.4

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

$f'' (x) = 5 (\frac{4 (x^{10} + 1) x^{3} - x^{4} (10 x^{9} + \frac{d}{d x} (1))}{{(x^{10} + 1)}^{2}})$

Step 2.3.5

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

$f'' (x) = 5 (\frac{4 (x^{10} + 1) x^{3} - x^{4} (10 x^{9} + 0)}{{(x^{10} + 1)}^{2}})$

Step 2.3.6

Simplify the expression.

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Step 2.3.6.1

Add $10 x^{9}$ and $0$ .

$f'' (x) = 5 (\frac{4 (x^{10} + 1) x^{3} - x^{4} (10 x^{9})}{{(x^{10} + 1)}^{2}})$

Step 2.3.6.2

Multiply $10$ by $- 1$ .

$f'' (x) = 5 (\frac{4 (x^{10} + 1) x^{3} - 10 x^{4} x^{9}}{{(x^{10} + 1)}^{2}})$

Step 2.4

Multiply $x^{4}$ by $x^{9}$ by adding the exponents.

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Step 2.4.1

Move $x^{9}$ .

$f'' (x) = 5 (\frac{4 (x^{10} + 1) x^{3} - 10 (x^{9} x^{4})}{{(x^{10} + 1)}^{2}})$

Step 2.4.2

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

$f'' (x) = 5 (\frac{4 (x^{10} + 1) x^{3} - 10 x^{9 + 4}}{{(x^{10} + 1)}^{2}})$

Step 2.4.3

Add $9$ and $4$ .

$f'' (x) = 5 (\frac{4 (x^{10} + 1) x^{3} - 10 x^{13}}{{(x^{10} + 1)}^{2}})$

Step 2.5

Combine $5$ and $\frac{4 (x^{10} + 1) x^{3} - 10 x^{13}}{{(x^{10} + 1)}^{2}}$ .

$f'' (x) = \frac{5 (4 (x^{10} + 1) x^{3} - 10 x^{13})}{{(x^{10} + 1)}^{2}}$

Step 2.6

Simplify.

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Step 2.6.1

Apply the distributive property.

$f'' (x) = \frac{5 ((4 x^{10} + 4 \cdot 1) x^{3} - 10 x^{13})}{{(x^{10} + 1)}^{2}}$

Step 2.6.2

Apply the distributive property.

$f'' (x) = \frac{5 (4 x^{10} x^{3} + 4 \cdot (1 x^{3}) - 10 x^{13})}{{(x^{10} + 1)}^{2}}$

Step 2.6.3

Apply the distributive property.

$f'' (x) = \frac{5 (4 x^{10} x^{3}) + 5 (4 \cdot (1 x^{3})) + 5 (- 10 x^{13})}{{(x^{10} + 1)}^{2}}$

Step 2.6.4

Simplify the numerator.

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Step 2.6.4.1

Simplify each term.

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Step 2.6.4.1.1

Multiply $x^{10}$ by $x^{3}$ by adding the exponents.

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Step 2.6.4.1.1.1

Move $x^{3}$ .

$f'' (x) = \frac{5 (4 (x^{3} x^{10})) + 5 (4 \cdot (1 x^{3})) + 5 (- 10 x^{13})}{{(x^{10} + 1)}^{2}}$

Step 2.6.4.1.1.2

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

$f'' (x) = \frac{5 (4 x^{3 + 10}) + 5 (4 \cdot (1 x^{3})) + 5 (- 10 x^{13})}{{(x^{10} + 1)}^{2}}$

Step 2.6.4.1.1.3

Add $3$ and $10$ .

$f'' (x) = \frac{5 (4 x^{13}) + 5 (4 \cdot (1 x^{3})) + 5 (- 10 x^{13})}{{(x^{10} + 1)}^{2}}$

Step 2.6.4.1.2

Multiply $4$ by $5$ .

$f'' (x) = \frac{20 x^{13} + 5 (4 \cdot (1 x^{3})) + 5 (- 10 x^{13})}{{(x^{10} + 1)}^{2}}$

Step 2.6.4.1.3

Multiply $4$ by $1$ .

$f'' (x) = \frac{20 x^{13} + 5 (4 x^{3}) + 5 (- 10 x^{13})}{{(x^{10} + 1)}^{2}}$

Step 2.6.4.1.4

Multiply $4$ by $5$ .

$f'' (x) = \frac{20 x^{13} + 20 x^{3} + 5 (- 10 x^{13})}{{(x^{10} + 1)}^{2}}$

Step 2.6.4.1.5

Multiply $- 10$ by $5$ .

$f'' (x) = \frac{20 x^{13} + 20 x^{3} - 50 x^{13}}{{(x^{10} + 1)}^{2}}$

Step 2.6.4.2

Subtract $50 x^{13}$ from $20 x^{13}$ .

$f'' (x) = \frac{- 30 x^{13} + 20 x^{3}}{{(x^{10} + 1)}^{2}}$

Step 2.6.5

Factor $10 x^{3}$ out of $- 30 x^{13} + 20 x^{3}$ .

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Step 2.6.5.1

Factor $10 x^{3}$ out of $- 30 x^{13}$ .

$f'' (x) = \frac{10 x^{3} (- 3 x^{10}) + 20 x^{3}}{{(x^{10} + 1)}^{2}}$

Step 2.6.5.2

Factor $10 x^{3}$ out of $20 x^{3}$ .

$f'' (x) = \frac{10 x^{3} (- 3 x^{10}) + 10 x^{3} (2)}{{(x^{10} + 1)}^{2}}$

Step 2.6.5.3

Factor $10 x^{3}$ out of $10 x^{3} (- 3 x^{10}) + 10 x^{3} (2)$ .

$f'' (x) = \frac{10 x^{3} (- 3 x^{10} + 2)}{{(x^{10} + 1)}^{2}}$

Step 2.6.6

Factor $- 1$ out of $- 3 x^{10}$ .

$f'' (x) = \frac{10 x^{3} (- (3 x^{10}) + 2)}{{(x^{10} + 1)}^{2}}$

Step 2.6.7

Rewrite $2$ as $- 1 (- 2)$ .

$f'' (x) = \frac{10 x^{3} (- (3 x^{10}) - 1 \cdot - 2)}{{(x^{10} + 1)}^{2}}$

Step 2.6.8

Factor $- 1$ out of $- (3 x^{10}) - 1 (- 2)$ .

$f'' (x) = \frac{10 x^{3} (- (3 x^{10} - 2))}{{(x^{10} + 1)}^{2}}$

Step 2.6.9

Rewrite $- (3 x^{10} - 2)$ as $- 1 (3 x^{10} - 2)$ .

$f'' (x) = \frac{10 x^{3} (- 1 (3 x^{10} - 2))}{{(x^{10} + 1)}^{2}}$

Step 2.6.10

Move the negative in front of the fraction.

$f'' (x) = - \frac{(10 x^{3}) (3 x^{10} - 2)}{{(x^{10} + 1)}^{2}}$

Step 2.6.11

Reorder factors in $- \frac{(10 x^{3}) (3 x^{10} - 2)}{{(x^{10} + 1)}^{2}}$ .

$f'' (x) = - \frac{10 x^{3} (3 x^{10} - 2)}{{(x^{10} + 1)}^{2}}$

Step 3

To find the local maximum and minimum values of the function, set the derivative equal to $0$ and solve.

$\frac{5 x^{4}}{x^{10} + 1} = 0$

Step 4

Set the numerator equal to zero.

$5 x^{4} = 0$

Step 5

Solve the equation for $x$ .

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Step 5.1

Divide each term in $5 x^{4} = 0$ by $5$ and simplify.

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Step 5.1.1

Divide each term in $5 x^{4} = 0$ by $5$ .

$\frac{5 x^{4}}{5} = \frac{0}{5}$

Step 5.1.2

Simplify the left side.

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Step 5.1.2.1

Cancel the common factor of $5$ .

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Step 5.1.2.1.1

Cancel the common factor.

$\frac{5 x^{4}}{5} = \frac{0}{5}$

Step 5.1.2.1.2

Divide $x^{4}$ by $1$ .

$x^{4} = \frac{0}{5}$

Step 5.1.3

Simplify the right side.

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Step 5.1.3.1

Divide $0$ by $5$ .

$x^{4} = 0$

Step 5.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt[4]{0}$

Step 5.3

Simplify $\pm \sqrt[4]{0}$ .

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Step 5.3.1

Rewrite $0$ as $0^{4}$ .

$x = \pm \sqrt[4]{0^{4}}$

Step 5.3.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 0$

Step 5.3.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 6

Evaluate the second derivative at $x = 0$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- \frac{10 {(0)}^{3} (3 {(0)}^{10} - 2)}{{({(0)}^{10} + 1)}^{2}}$

Step 7

Evaluate the second derivative.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Raising $0$ to any positive power yields $0$ .

$- \frac{10 \cdot 0^{3} (3 \cdot 0 - 2)}{{(0^{10} + 1)}^{2}}$

Step 7.1.2

Multiply $3$ by $0$ .

$- \frac{10 \cdot 0^{3} (0 - 2)}{{(0^{10} + 1)}^{2}}$

Step 7.1.3

Subtract $2$ from $0$ .

$- \frac{10 \cdot 0^{3} \cdot - 2}{{(0^{10} + 1)}^{2}}$

Step 7.1.4

Multiply $- 2$ by $10$ .

$- \frac{- 20 \cdot 0^{3}}{{(0^{10} + 1)}^{2}}$

Step 7.1.5

Raising $0$ to any positive power yields $0$ .

$- \frac{- 20 \cdot 0}{{(0^{10} + 1)}^{2}}$

Step 7.2

Simplify the denominator.

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Step 7.2.1

Raising $0$ to any positive power yields $0$ .

$- \frac{- 20 \cdot 0}{{(0 + 1)}^{2}}$

Step 7.2.2

Add $0$ and $1$ .

$- \frac{- 20 \cdot 0}{1^{2}}$

Step 7.2.3

One to any power is one.

$- \frac{- 20 \cdot 0}{1}$

Step 7.3

Simplify the expression.

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Step 7.3.1

Multiply $- 20$ by $0$ .

$- \frac{0}{1}$

Step 7.3.2

Divide $0$ by $1$ .

$- 0$

Step 7.3.3

Multiply $- 1$ by $0$ .

$0$

Step 8

Since there is at least one point with $0$ or undefined second derivative, apply the first derivative test.

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Step 8.1

Split $(- \infty, \infty)$ into separate intervals around the $x$ values that make the first derivative $0$ or undefined.

$(- \infty, 0) \cup (0, \infty)$

Step 8.2

Substitute any number, such as $- 2$ , from the interval $(- \infty, 0)$ in the first derivative $\frac{5 x^{4}}{x^{10} + 1}$ to check if the result is negative or positive.

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Step 8.2.1

Replace the variable $x$ with $- 2$ in the expression.

$f' (- 2) = \frac{5 {(- 2)}^{4}}{{(- 2)}^{10} + 1}$

Step 8.2.2

Simplify the result.

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Step 8.2.2.1

Raise $- 2$ to the power of $4$ .

$f' (- 2) = \frac{5 \cdot 16}{{(- 2)}^{10} + 1}$

Step 8.2.2.2

Simplify the denominator.

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Step 8.2.2.2.1

Raise $- 2$ to the power of $10$ .

$f' (- 2) = \frac{5 \cdot 16}{1024 + 1}$

Step 8.2.2.2.2

Add $1024$ and $1$ .

$f' (- 2) = \frac{5 \cdot 16}{1025}$

Step 8.2.2.3

Reduce the expression by cancelling the common factors.

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Step 8.2.2.3.1

Multiply $5$ by $16$ .

$f' (- 2) = \frac{80}{1025}$

Step 8.2.2.3.2

Cancel the common factor of $80$ and $1025$ .

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Step 8.2.2.3.2.1

Factor $5$ out of $80$ .

$f' (- 2) = \frac{5 (16)}{1025}$

Step 8.2.2.3.2.2

Cancel the common factors.

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Step 8.2.2.3.2.2.1

Factor $5$ out of $1025$ .

$f' (- 2) = \frac{5 \cdot 16}{5 \cdot 205}$

Step 8.2.2.3.2.2.2

Cancel the common factor.

$f' (- 2) = \frac{5 \cdot 16}{5 \cdot 205}$

Step 8.2.2.3.2.2.3

Rewrite the expression.

$f' (- 2) = \frac{16}{205}$

Step 8.2.2.4

The final answer is $\frac{16}{205}$ .

$\frac{16}{205}$

Step 8.3

Substitute any number, such as $2$ , from the interval $(0, \infty)$ in the first derivative $\frac{5 x^{4}}{x^{10} + 1}$ to check if the result is negative or positive.

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Step 8.3.1

Replace the variable $x$ with $2$ in the expression.

$f' (2) = \frac{5 {(2)}^{4}}{{(2)}^{10} + 1}$

Step 8.3.2

Simplify the result.

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Step 8.3.2.1

Raise $2$ to the power of $4$ .

$f' (2) = \frac{5 \cdot 16}{2^{10} + 1}$

Step 8.3.2.2

Simplify the denominator.

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Step 8.3.2.2.1

Raise $2$ to the power of $10$ .

$f' (2) = \frac{5 \cdot 16}{1024 + 1}$

Step 8.3.2.2.2

Add $1024$ and $1$ .

$f' (2) = \frac{5 \cdot 16}{1025}$

Step 8.3.2.3

Reduce the expression by cancelling the common factors.

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Step 8.3.2.3.1

Multiply $5$ by $16$ .

$f' (2) = \frac{80}{1025}$

Step 8.3.2.3.2

Cancel the common factor of $80$ and $1025$ .

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Step 8.3.2.3.2.1

Factor $5$ out of $80$ .

$f' (2) = \frac{5 (16)}{1025}$

Step 8.3.2.3.2.2

Cancel the common factors.

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Step 8.3.2.3.2.2.1

Factor $5$ out of $1025$ .

$f' (2) = \frac{5 \cdot 16}{5 \cdot 205}$

Step 8.3.2.3.2.2.2

Cancel the common factor.

$f' (2) = \frac{5 \cdot 16}{5 \cdot 205}$

Step 8.3.2.3.2.2.3

Rewrite the expression.

$f' (2) = \frac{16}{205}$

Step 8.3.2.4

The final answer is $\frac{16}{205}$ .

$\frac{16}{205}$

Step 8.4

Since the first derivative did not change signs around $x = 0$ , this is not a local maximum or minimum.

Not a local maximum or minimum

Step 8.5

No local maxima or minima found for $f (x) = \arctan (x^{5})$ .

No local maxima or minima

Step 9