Calculus Examples

$f (x) = x^{4} - 4 x^{3} + 10$

Step 1

Find the first derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

Differentiate.

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Step 1.1.1.1

By the Sum Rule, the derivative of $x^{4} - 4 x^{3} + 10$ with respect to $x$ is $\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 4 x^{3}] + \frac{d}{d x} [10]$ .

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

Step 1.1.1.2

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

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

Step 1.1.2

Evaluate $\frac{d}{d x} [- 4 x^{3}]$ .

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Step 1.1.2.1

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4 x^{3}$ with respect to $x$ is $- 4 \frac{d}{d x} [x^{3}]$ .

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

Step 1.1.2.2

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

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

Step 1.1.2.3

Multiply $3$ by $- 4$ .

$4 x^{3} - 12 x^{2} + \frac{d}{d x} [10]$

Step 1.1.3

Differentiate using the Constant Rule.

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Step 1.1.3.1

Since $10$ is constant with respect to $x$ , the derivative of $10$ with respect to $x$ is $0$ .

$4 x^{3} - 12 x^{2} + 0$

Step 1.1.3.2

Add $4 x^{3} - 12 x^{2}$ and $0$ .

$f' (x) = 4 x^{3} - 12 x^{2}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $4 x^{3} - 12 x^{2}$ .

$4 x^{3} - 12 x^{2}$

Step 2

Set the first derivative equal to $0$ then solve the equation $4 x^{3} - 12 x^{2} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$4 x^{3} - 12 x^{2} = 0$

Step 2.2

Factor $4 x^{2}$ out of $4 x^{3} - 12 x^{2}$ .

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Step 2.2.1

Factor $4 x^{2}$ out of $4 x^{3}$ .

$4 x^{2} (x) - 12 x^{2} = 0$

Step 2.2.2

Factor $4 x^{2}$ out of $- 12 x^{2}$ .

$4 x^{2} (x) + 4 x^{2} (- 3) = 0$

Step 2.2.3

Factor $4 x^{2}$ out of $4 x^{2} (x) + 4 x^{2} (- 3)$ .

$4 x^{2} (x - 3) = 0$

Step 2.3

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

$x^{2} = 0$

$x - 3 = 0$

Step 2.4

Set $x^{2}$ equal to $0$ and solve for $x$ .

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Step 2.4.1

Set $x^{2}$ equal to $0$ .

$x^{2} = 0$

Step 2.4.2

Solve $x^{2} = 0$ for $x$ .

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Step 2.4.2.1

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

$x = \pm \sqrt{0}$

Step 2.4.2.2

Simplify $\pm \sqrt{0}$ .

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Step 2.4.2.2.1

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Step 2.4.2.2.2

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

$x = \pm 0$

Step 2.4.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 2.5

Set $x - 3$ equal to $0$ and solve for $x$ .

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Step 2.5.1

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 2.5.2

Add $3$ to both sides of the equation.

$x = 3$

Step 2.6

The final solution is all the values that make $4 x^{2} (x - 3) = 0$ true.

$x = 0, 3$

Step 3

The values which make the derivative equal to $0$ are $0, 3$ .

$0, 3$

Step 4

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

$(- \infty, 0) \cup (0, 3) \cup (3, \infty)$

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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Step 5.1

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

$f' (- 1) = 4 {(- 1)}^{3} - 12 {(- 1)}^{2}$

Step 5.2

Simplify the result.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

Raise $- 1$ to the power of $3$ .

$f' (- 1) = 4 \cdot - 1 - 12 {(- 1)}^{2}$

Step 5.2.1.2

Multiply $4$ by $- 1$ .

$f' (- 1) = - 4 - 12 {(- 1)}^{2}$

Step 5.2.1.3

Raise $- 1$ to the power of $2$ .

$f' (- 1) = - 4 - 12 \cdot 1$

Step 5.2.1.4

Multiply $- 12$ by $1$ .

$f' (- 1) = - 4 - 12$

Step 5.2.2

Subtract $12$ from $- 4$ .

$f' (- 1) = - 16$

Step 5.2.3

The final answer is $- 16$ .

$- 16$

Step 5.3

At $x = - 1$ the derivative is $- 16$ . 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 $(0, 3)$ into the derivative to determine if the function is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $\frac{3}{2}$ in the expression.

$f' (\frac{3}{2}) = 4 {(\frac{3}{2})}^{3} - 12 {(\frac{3}{2})}^{2}$

Step 6.2

Simplify the result.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

Apply the product rule to $\frac{3}{2}$ .

$f' (\frac{3}{2}) = 4 (\frac{3^{3}}{2^{3}}) - 12 {(\frac{3}{2})}^{2}$

Step 6.2.1.2

Raise $3$ to the power of $3$ .

$f' (\frac{3}{2}) = 4 (\frac{27}{2^{3}}) - 12 {(\frac{3}{2})}^{2}$

Step 6.2.1.3

Raise $2$ to the power of $3$ .

$f' (\frac{3}{2}) = 4 (\frac{27}{8}) - 12 {(\frac{3}{2})}^{2}$

Step 6.2.1.4

Cancel the common factor of $4$ .

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Step 6.2.1.4.1

Factor $4$ out of $8$ .

$f' (\frac{3}{2}) = 4 (\frac{27}{4 (2)}) - 12 {(\frac{3}{2})}^{2}$

Step 6.2.1.4.2

Cancel the common factor.

$f' (\frac{3}{2}) = 4 (\frac{27}{4 \cdot 2}) - 12 {(\frac{3}{2})}^{2}$

Step 6.2.1.4.3

Rewrite the expression.

$f' (\frac{3}{2}) = \frac{27}{2} - 12 {(\frac{3}{2})}^{2}$

Step 6.2.1.5

Apply the product rule to $\frac{3}{2}$ .

$f' (\frac{3}{2}) = \frac{27}{2} - 12 \frac{3^{2}}{2^{2}}$

Step 6.2.1.6

Raise $3$ to the power of $2$ .

$f' (\frac{3}{2}) = \frac{27}{2} - 12 \frac{9}{2^{2}}$

Step 6.2.1.7

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

$f' (\frac{3}{2}) = \frac{27}{2} - 12 (\frac{9}{4})$

Step 6.2.1.8

Cancel the common factor of $4$ .

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Step 6.2.1.8.1

Factor $4$ out of $- 12$ .

$f' (\frac{3}{2}) = \frac{27}{2} + 4 (- 3) (\frac{9}{4})$

Step 6.2.1.8.2

Cancel the common factor.

$f' (\frac{3}{2}) = \frac{27}{2} + 4 \cdot (- 3 (\frac{9}{4}))$

Step 6.2.1.8.3

Rewrite the expression.

$f' (\frac{3}{2}) = \frac{27}{2} - 3 \cdot 9$

Step 6.2.1.9

Multiply $- 3$ by $9$ .

$f' (\frac{3}{2}) = \frac{27}{2} - 27$

Step 6.2.2

To write $- 27$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f' (\frac{3}{2}) = \frac{27}{2} - 27 \cdot \frac{2}{2}$

Step 6.2.3

Combine $- 27$ and $\frac{2}{2}$ .

$f' (\frac{3}{2}) = \frac{27}{2} + \frac{- 27 \cdot 2}{2}$

Step 6.2.4

Combine the numerators over the common denominator.

$f' (\frac{3}{2}) = \frac{27 - 27 \cdot 2}{2}$

Step 6.2.5

Simplify the numerator.

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Step 6.2.5.1

Multiply $- 27$ by $2$ .

$f' (\frac{3}{2}) = \frac{27 - 54}{2}$

Step 6.2.5.2

Subtract $54$ from $27$ .

$f' (\frac{3}{2}) = \frac{- 27}{2}$

Step 6.2.6

Move the negative in front of the fraction.

$f' (\frac{3}{2}) = - \frac{27}{2}$

Step 6.2.7

The final answer is $- \frac{27}{2}$ .

$- \frac{27}{2}$

Step 6.3

At $x = \frac{3}{2}$ the derivative is $- \frac{27}{2}$ . Since this is negative, the function is decreasing on $(0, 3)$ .

Decreasing on $(0, 3)$ since $f' (x) < 0$

Step 7

Substitute a value from the interval $(3, \infty)$ into the derivative to determine if the function is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $4$ in the expression.

$f' (4) = 4 {(4)}^{3} - 12 {(4)}^{2}$

Step 7.2

Simplify the result.

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Step 7.2.1

Simplify each term.

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Step 7.2.1.1

Multiply $4$ by ${(4)}^{3}$ by adding the exponents.

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Step 7.2.1.1.1

Multiply $4$ by ${(4)}^{3}$ .

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Step 7.2.1.1.1.1

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

$f' (4) = 4 {(4)}^{3} - 12 {(4)}^{2}$

Step 7.2.1.1.1.2

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

$f' (4) = 4^{1 + 3} - 12 {(4)}^{2}$

Step 7.2.1.1.2

Add $1$ and $3$ .

$f' (4) = 4^{4} - 12 {(4)}^{2}$

Step 7.2.1.2

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

$f' (4) = 256 - 12 {(4)}^{2}$

Step 7.2.1.3

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

$f' (4) = 256 - 12 \cdot 16$

Step 7.2.1.4

Multiply $- 12$ by $16$ .

$f' (4) = 256 - 192$

Step 7.2.2

Subtract $192$ from $256$ .

$f' (4) = 64$

Step 7.2.3

The final answer is $64$ .

$64$

Step 7.3

At $x = 4$ the derivative is $64$ . Since this is positive, the function is increasing on $(3, \infty)$ .

Increasing on $(3, \infty)$ since $f' (x) > 0$

Step 8

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

Increasing on: $(3, \infty)$

Decreasing on: $(- \infty, 0), (0, 3)$

Step 9