Calculus Examples

$f (x)$

Step 1

Find the first derivative.

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Step 1.1

Differentiate using the function rule which states that the derivative of $f (x)$ is $\frac{d f}{d x}$ .

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

Step 1.2

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

$\frac{d f}{d x}$

Step 2

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

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Step 2.1

Set the first derivative equal to $0$ .

$\frac{d f}{d x} = 0$

Step 2.2

Cancel the common factor of $d$ .

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Step 2.2.1

Cancel the common factor.

$\frac{d f}{d x} = 0$

Step 2.2.2

Rewrite the expression.

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

Step 2.3

Multiply both sides of the equation by $x$ .

$f = x (0)$

Step 2.4

Rewrite the equation as $x (0) = f$ .

$x (0) = f$

Step 2.5

Multiply $x$ by $0$ .

$0 = f$

Step 2.6

Rewrite $0 = f$ so $f$ is on the left side.

$f = 0$

Step 2.7

The variable $x$ got canceled.

All real numbers

Step 3

Find the values where the derivative is undefined.

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Step 3.1

Set the denominator in $\frac{d f}{d x}$ equal to $0$ to find where the expression is undefined.

$d x = 0$

Step 3.2

Divide each term in $d x = 0$ by $x$ and simplify.

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Step 3.2.1

Divide each term in $d x = 0$ by $x$ .

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

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $x$ .

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Step 3.2.2.1.1

Cancel the common factor.

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

Step 3.2.2.1.2

Divide $d$ by $1$ .

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

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Divide $0$ by $x$ .

$d = 0$

Step 4

Evaluate $f (x)$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = A l \cdot l (r e a l) (n u m b e r s)$ .

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Step 4.1.1

Substitute All real numbers for $x$ .

$f (A l \cdot l (r e a l) (n u m b e r s))$

Step 4.1.2

Simplify.

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Step 4.1.2.1

Rewrite using the commutative property of multiplication.

$e e f A l \cdot l r a l n u m b r s$

Step 4.1.2.2

Multiply $l$ by $l$ by adding the exponents.

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Step 4.1.2.2.1

Move $l$ .

$e e f A (l \cdot l) r a l n u m b r s$

Step 4.1.2.2.2

Multiply $l$ by $l$ .

$e e f A l^{2} r a l n u m b r s$

Step 4.1.2.3

Multiply $l^{2}$ by $l$ by adding the exponents.

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Step 4.1.2.3.1

Move $l$ .

$e e f A (l \cdot l^{2}) r a n u m b r s$

Step 4.1.2.3.2

Multiply $l$ by $l^{2}$ .

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Step 4.1.2.3.2.1

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

$e e f A (l^{1} l^{2}) r a n u m b r s$

Step 4.1.2.3.2.2

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

$e e f A l^{1 + 2} r a n u m b r s$

Step 4.1.2.3.3

Add $1$ and $2$ .

$e e f A l^{3} r a n u m b r s$

Step 4.1.2.4

Multiply $r$ by $r$ by adding the exponents.

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Step 4.1.2.4.1

Move $r$ .

$e e f A l^{3} (r \cdot r) a n u m b s$

Step 4.1.2.4.2

Multiply $r$ by $r$ .

$e e f A l^{3} r^{2} a n u m b s$

Step 4.1.2.5

Multiply $e e$ .

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Step 4.1.2.5.1

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

$e^{1} e f A l^{3} r^{2} a n u m b s$

Step 4.1.2.5.2

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

$e^{1} e^{1} f A l^{3} r^{2} a n u m b s$

Step 4.1.2.5.3

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

$e^{1 + 1} f A l^{3} r^{2} a n u m b s$

Step 4.1.2.5.4

Add $1$ and $1$ .

$e^{2} f A l^{3} r^{2} a n u m b s$

Step 4.2

Evaluate at $x = 0$ .

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Step 4.2.1

Substitute $0$ for $x$ .

$f (0)$

Step 4.2.2

Multiply $f$ by $0$ .

$0$

Step 4.3

List all of the points.

$(A l \cdot l (r e a l) (n u m b e r s), e^{2} f A l^{3} r^{2} a n u m b s), (0, 0)$ , for any integer $n$

Step 5