Calculus Examples

$f (x) = 2 {(\frac{x}{4})}^{2}$

Step 1

Find the first derivative of the function.

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Step 1.1

Simplify terms.

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Step 1.1.1

Apply the product rule to $\frac{x}{4}$ .

$\frac{d}{d x} [2 \frac{x^{2}}{4^{2}}]$

Step 1.1.2

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

$\frac{d}{d x} [2 \frac{x^{2}}{16}]$

Step 1.1.3

Combine $2$ and $\frac{x^{2}}{16}$ .

$\frac{d}{d x} [\frac{2 x^{2}}{16}]$

Step 1.1.4

Cancel the common factor of $2$ and $16$ .

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Step 1.1.4.1

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

$\frac{d}{d x} [\frac{2 (x^{2})}{16}]$

Step 1.1.4.2

Cancel the common factors.

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Step 1.1.4.2.1

Factor $2$ out of $16$ .

$\frac{d}{d x} [\frac{2 x^{2}}{2 \cdot 8}]$

Step 1.1.4.2.2

Cancel the common factor.

$\frac{d}{d x} [\frac{2 x^{2}}{2 \cdot 8}]$

Step 1.1.4.2.3

Rewrite the expression.

$\frac{d}{d x} [\frac{x^{2}}{8}]$

Step 1.2

Since $\frac{1}{8}$ is constant with respect to $x$ , the derivative of $\frac{x^{2}}{8}$ with respect to $x$ is $\frac{1}{8} \frac{d}{d x} [x^{2}]$ .

$\frac{1}{8} \frac{d}{d x} [x^{2}]$

Step 1.3

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

$\frac{1}{8} (2 x)$

Step 1.4

Simplify terms.

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Step 1.4.1

Combine $2$ and $\frac{1}{8}$ .

$\frac{2}{8} x$

Step 1.4.2

Combine $\frac{2}{8}$ and $x$ .

$\frac{2 x}{8}$

Step 1.4.3

Cancel the common factor of $2$ and $8$ .

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Step 1.4.3.1

Factor $2$ out of $2 x$ .

$\frac{2 (x)}{8}$

Step 1.4.3.2

Cancel the common factors.

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Step 1.4.3.2.1

Factor $2$ out of $8$ .

$\frac{2 x}{2 \cdot 4}$

Step 1.4.3.2.2

Cancel the common factor.

$\frac{2 x}{2 \cdot 4}$

Step 1.4.3.2.3

Rewrite the expression.

$\frac{x}{4}$

Step 2

Find the second derivative of the function.

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Step 2.1

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

$f'' (x) = \frac{1}{4} \cdot \frac{d}{d x} (x)$

Step 2.2

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

$f'' (x) = \frac{1}{4} \cdot 1$

Step 2.3

Multiply $\frac{1}{4}$ by $1$ .

$f'' (x) = \frac{1}{4}$

Step 3

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

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

Step 4

Find the first derivative.

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Step 4.1

Find the first derivative.

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Step 4.1.1

Simplify terms.

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Step 4.1.1.1

Apply the product rule to $\frac{x}{4}$ .

$\frac{d}{d x} [2 \frac{x^{2}}{4^{2}}]$

Step 4.1.1.2

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

$\frac{d}{d x} [2 \frac{x^{2}}{16}]$

Step 4.1.1.3

Combine $2$ and $\frac{x^{2}}{16}$ .

$\frac{d}{d x} [\frac{2 x^{2}}{16}]$

Step 4.1.1.4

Cancel the common factor of $2$ and $16$ .

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Step 4.1.1.4.1

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

$\frac{d}{d x} [\frac{2 (x^{2})}{16}]$

Step 4.1.1.4.2

Cancel the common factors.

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Step 4.1.1.4.2.1

Factor $2$ out of $16$ .

$\frac{d}{d x} [\frac{2 x^{2}}{2 \cdot 8}]$

Step 4.1.1.4.2.2

Cancel the common factor.

$\frac{d}{d x} [\frac{2 x^{2}}{2 \cdot 8}]$

Step 4.1.1.4.2.3

Rewrite the expression.

$\frac{d}{d x} [\frac{x^{2}}{8}]$

Step 4.1.2

Since $\frac{1}{8}$ is constant with respect to $x$ , the derivative of $\frac{x^{2}}{8}$ with respect to $x$ is $\frac{1}{8} \frac{d}{d x} [x^{2}]$ .

$\frac{1}{8} \frac{d}{d x} [x^{2}]$

Step 4.1.3

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

$\frac{1}{8} (2 x)$

Step 4.1.4

Simplify terms.

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Step 4.1.4.1

Combine $2$ and $\frac{1}{8}$ .

$\frac{2}{8} x$

Step 4.1.4.2

Combine $\frac{2}{8}$ and $x$ .

$\frac{2 x}{8}$

Step 4.1.4.3

Cancel the common factor of $2$ and $8$ .

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Step 4.1.4.3.1

Factor $2$ out of $2 x$ .

$\frac{2 (x)}{8}$

Step 4.1.4.3.2

Cancel the common factors.

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Step 4.1.4.3.2.1

Factor $2$ out of $8$ .

$\frac{2 x}{2 \cdot 4}$

Step 4.1.4.3.2.2

Cancel the common factor.

$\frac{2 x}{2 \cdot 4}$

Step 4.1.4.3.2.3

Rewrite the expression.

$f' (x) = \frac{x}{4}$

Step 4.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{x}{4}$ .

$\frac{x}{4}$

Step 5

Set the first derivative equal to $0$ then solve the equation $\frac{x}{4} = 0$ .

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Step 5.1

Set the first derivative equal to $0$ .

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

Step 5.2

Set the numerator equal to zero.

$x = 0$

Step 6

Find the values where the derivative is undefined.

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Step 6.1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 7

Critical points to evaluate.

$x = 0$

Step 8

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.

$\frac{1}{4}$

Step 9

$x = 0$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = 0$ is a local minimum

Step 10

Find the y-value when $x = 0$ .

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Step 10.1

Replace the variable $x$ with $0$ in the expression.

$f (0) = 2 {(\frac{0}{4})}^{2}$

Step 10.2

Simplify the result.

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Step 10.2.1

Divide $0$ by $4$ .

$f (0) = 2 \cdot 0^{2}$

Step 10.2.2

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

$f (0) = 2 \cdot 0$

Step 10.2.3

Multiply $2$ by $0$ .

$f (0) = 0$

Step 10.2.4

The final answer is $0$ .

$y = 0$

Step 11

These are the local extrema for $f (x) = 2 {(\frac{x}{4})}^{2}$ .

$(0, 0)$ is a local minima

Step 12