Calculus Examples

$f (x) = 4 x^{6} - x^{5} - 24$

Step 1

Find the first derivative.

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Find the first derivative.

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By the Sum Rule, the derivative of $4 x^{6} - x^{5} - 24$ with respect to $x$ is $\frac{d}{d x} [4 x^{6}] + \frac{d}{d x} [- x^{5}] + \frac{d}{d x} [- 24]$ .

$f' (x) = \frac{d}{d x} (4 x^{6}) + \frac{d}{d x} (- x^{5}) + \frac{d}{d x} (- 24)$

Evaluate $\frac{d}{d x} [4 x^{6}]$ .

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Since $4$ is constant with respect to $x$ , the derivative of $4 x^{6}$ with respect to $x$ is $4 \frac{d}{d x} [x^{6}]$ .

$f' (x) = 4 \frac{d}{d x} (x^{6}) + \frac{d}{d x} (- x^{5}) + \frac{d}{d x} (- 24)$

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

$f' (x) = 4 (6 x^{5}) + \frac{d}{d x} (- x^{5}) + \frac{d}{d x} (- 24)$

Multiply $6$ by $4$ .

$f' (x) = 24 x^{5} + \frac{d}{d x} (- x^{5}) + \frac{d}{d x} (- 24)$

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

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

$f' (x) = 24 x^{5} - \frac{d}{d x} x^{5} + \frac{d}{d x} (- 24)$

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

$f' (x) = 24 x^{5} - (5 x^{4}) + \frac{d}{d x} (- 24)$

Multiply $5$ by $- 1$ .

$f' (x) = 24 x^{5} - 5 x^{4} + \frac{d}{d x} (- 24)$

Differentiate using the Constant Rule.

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Since $- 24$ is constant with respect to $x$ , the derivative of $- 24$ with respect to $x$ is $0$ .

$f' (x) = 24 x^{5} - 5 x^{4} + 0$

Add $24 x^{5} - 5 x^{4}$ and $0$ .

$f' (x) = 24 x^{5} - 5 x^{4}$

The first derivative of $f (x)$ with respect to $x$ is $24 x^{5} - 5 x^{4}$ .

$24 x^{5} - 5 x^{4}$

Step 2

Set the first derivative equal to $0$ then solve the equation $24 x^{5} - 5 x^{4} = 0$ .

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Set the first derivative equal to $0$ .

$24 x^{5} - 5 x^{4} = 0$

Factor $x^{4}$ out of $24 x^{5} - 5 x^{4}$ .

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Factor $x^{4}$ out of $24 x^{5}$ .

$x^{4} (24 x) - 5 x^{4} = 0$

Factor $x^{4}$ out of $- 5 x^{4}$ .

$x^{4} (24 x) + x^{4} \cdot - 5 = 0$

Factor $x^{4}$ out of $x^{4} (24 x) + x^{4} \cdot - 5$ .

$x^{4} (24 x - 5) = 0$

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x^{4} = 0$

$24 x - 5 = 0$

Set $x^{4}$ equal to $0$ and solve for $x$ .

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Set $x^{4}$ equal to $0$ .

$x^{4} = 0$

Solve $x^{4} = 0$ for $x$ .

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Take the 4th root of both sides of the equation to eliminate the exponent on the left side.

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

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

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

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

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

$x = \pm 0$

Plus or minus $0$ is $0$ .

$x = 0$

Set $24 x - 5$ equal to $0$ and solve for $x$ .

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Set $24 x - 5$ equal to $0$ .

$24 x - 5 = 0$

Solve $24 x - 5 = 0$ for $x$ .

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Add $5$ to both sides of the equation.

$24 x = 5$

Divide each term in $24 x = 5$ by $24$ and simplify.

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Divide each term in $24 x = 5$ by $24$ .

$\frac{24 x}{24} = \frac{5}{24}$

Simplify the left side.

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Cancel the common factor of $24$ .

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Cancel the common factor.

$\frac{24 x}{24} = \frac{5}{24}$

Divide $x$ by $1$ .

$x = \frac{5}{24}$

The final solution is all the values that make $x^{4} (24 x - 5) = 0$ true.

$x = 0, \frac{5}{24}$

Step 3

The values which make the derivative equal to $0$ are $0, \frac{5}{24}$ .

$0, \frac{5}{24}$

Step 4

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

$(- \infty, 0) \cup (0, \frac{5}{24}) \cup (\frac{5}{24}, \infty)$

Step 5

Substitute a value from the interval $(- \infty, 0)$ into the derivative to determine if the function is increasing or decreasing.

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Replace the variable $x$ with $- 1$ in the expression.

$f' (- 1) = 24 {(- 1)}^{5} - 5 {(- 1)}^{4}$

Simplify the result.

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Simplify each term.

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Raise $- 1$ to the power of $5$ .

$f' (- 1) = 24 \cdot - 1 - 5 {(- 1)}^{4}$

Multiply $24$ by $- 1$ .

$f' (- 1) = - 24 - 5 {(- 1)}^{4}$

Raise $- 1$ to the power of $4$ .

$f' (- 1) = - 24 - 5 \cdot 1$

Multiply $- 5$ by $1$ .

$f' (- 1) = - 24 - 5$

Subtract $5$ from $- 24$ .

$f' (- 1) = - 29$

The final answer is $- 29$ .

$- 29$

At $x = - 1$ the derivative is $- 29$ . Since this is negative, the function is decreasing on $(- \infty, 0)$ .

Decreasing on $(- \infty, 0)$ since $f' (x) < 0$

Step 6

Substitute a value from the interval $(0, \frac{5}{24})$ into the derivative to determine if the function is increasing or decreasing.

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Replace the variable $x$ with $0.1041\overline{6}$ in the expression.

$f' (0.1041\overline{6}) = 24 {(0.1041\overline{6})}^{5} - 5 {(0.1041\overline{6})}^{4}$

Simplify the result.

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Simplify each term.

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Raise $0.1041\overline{6}$ to the power of $5$ .

$f' (0.1041\overline{6}) = 24 \cdot 0.00001226 - 5 {(0.1041\overline{6})}^{4}$

Multiply $24$ by $0.00001226$ .

$f' (0.1041\overline{6}) = 0.00029434 - 5 {(0.1041\overline{6})}^{4}$

Raise $0.1041\overline{6}$ to the power of $4$ .

$f' (0.1041\overline{6}) = 0.00029434 - 5 \cdot 0.00011773$

Multiply $- 5$ by $0.00011773$ .

$f' (0.1041\overline{6}) = 0.00029434 - 0.00058868$

Subtract $0.00058868$ from $0.00029434$ .

$f' (0.1041\overline{6}) = - 0.00029434$

The final answer is $- 0.00029434$ .

$- 0.00029434$

At $x = 0.1041\overline{6}$ the derivative is $- 0.00029434$ . Since this is negative, the function is decreasing on $(0, \frac{5}{24})$ .

Decreasing on $(0, \frac{5}{24})$ since $f' (x) < 0$

Step 7

Substitute a value from the interval $(\frac{5}{24}, \infty)$ into the derivative to determine if the function is increasing or decreasing.

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Replace the variable $x$ with $1.208\overline{3}$ in the expression.

$f' (1.208\overline{3}) = 24 {(1.208\overline{3})}^{5} - 5 {(1.208\overline{3})}^{4}$

Simplify the result.

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Simplify each term.

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Raise $1.208\overline{3}$ to the power of $5$ .

$f' (1.208\overline{3}) = 24 \cdot 2.57592832 - 5 {(1.208\overline{3})}^{4}$

Multiply $24$ by $2.57592832$ .

$f' (1.208\overline{3}) = 61.82227984 - 5 {(1.208\overline{3})}^{4}$

Raise $1.208\overline{3}$ to the power of $4$ .

$f' (1.208\overline{3}) = 61.82227984 - 5 \cdot 2.13180275$

Multiply $- 5$ by $2.13180275$ .

$f' (1.208\overline{3}) = 61.82227984 - 10.65901379$

Subtract $10.65901379$ from $61.82227984$ .

$f' (1.208\overline{3}) = 51.16326604$

The final answer is $51.16326604$ .

$51.16326604$

At $x = 1.208\overline{3}$ the derivative is $51.16326604$ . Since this is positive, the function is increasing on $(\frac{5}{24}, \infty)$ .

Increasing on $(\frac{5}{24}, \infty)$ since $f' (x) > 0$

Step 8

List the intervals on which the function is increasing and decreasing.

Increasing on: $(\frac{5}{24}, \infty)$

Decreasing on: $(- \infty, 0), (0, \frac{5}{24})$

Step 9