Calculus Examples

$y = f (x)$

Step 1

Write $y = f (x)$ as a function.

$f (x) = f (x)$

Step 2

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

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

Step 3

Find the second derivative of the function.

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Step 3.1

Cancel the common factor of $d$ .

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Step 3.1.1

Cancel the common factor.

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

Step 3.1.2

Rewrite the expression.

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

Step 3.2

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

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

Step 3.3

Rewrite $\frac{1}{x}$ as $x^{- 1}$ .

$f'' (x) = f \frac{d}{d x} (x^{- 1})$

Step 3.4

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

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

Step 3.5

Simplify.

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Step 3.5.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 3.5.2

Combine $f$ and $\frac{1}{x^{2}}$ .

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

Step 4

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

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

Step 5

Find the first derivative.

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Step 5.1

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 5.2

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

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

Step 6

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

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Step 6.1

Set the first derivative equal to $0$ .

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

Step 6.2

Cancel the common factor of $d$ .

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Step 6.2.1

Cancel the common factor.

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

Step 6.2.2

Rewrite the expression.

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

Step 6.3

Multiply both sides of the equation by $x$ .

$f = x (0)$

Step 6.4

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

$x (0) = f$

Step 6.5

Multiply $x$ by $0$ .

$0 = f$

Step 6.6

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

$f = 0$

Step 6.7

The variable $x$ got canceled.

All real numbers

Step 7

Find the values where the derivative is undefined.

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Step 7.1

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

$d x = 0$

Step 7.2

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

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Step 7.2.1

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

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

Step 7.2.2

Simplify the left side.

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Step 7.2.2.1

Cancel the common factor of $x$ .

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Step 7.2.2.1.1

Cancel the common factor.

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

Step 7.2.2.1.2

Divide $d$ by $1$ .

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

Step 7.2.3

Simplify the right side.

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Step 7.2.3.1

Divide $0$ by $x$ .

$d = 0$

Step 8

Critical points to evaluate.

$x = A l \cdot l$ real $n u m b e r s, 0$

Step 9

Evaluate the second derivative at $x = A l \cdot l (r e a l) (n u m b e r s)$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

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

Step 10

Simplify the denominator.

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Step 10.1

Combine exponents.

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Step 10.1.1

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

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

Step 10.1.2

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

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

Step 10.1.3

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

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

Step 10.1.4

Add $1$ and $1$ .

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

Step 10.1.5

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

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

Step 10.1.6

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

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

Step 10.1.7

Add $1$ and $2$ .

$- \frac{f}{{(A l^{3} r e a n u m b e r s)}^{2}}$

Step 10.1.8

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

$- \frac{f}{{(A l^{3} r a n u m b (e^{1} e) r s)}^{2}}$

Step 10.1.9

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

$- \frac{f}{{(A l^{3} r a n u m b (e^{1} e^{1}) r s)}^{2}}$

Step 10.1.10

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

$- \frac{f}{{(A l^{3} r a n u m b e^{1 + 1} r s)}^{2}}$

Step 10.1.11

Add $1$ and $1$ .

$- \frac{f}{{(A l^{3} r a n u m b e^{2} r s)}^{2}}$

Step 10.1.12

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

$- \frac{f}{{(A l^{3} a n u m b e^{2} (r^{1} r) s)}^{2}}$

Step 10.1.13

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

$- \frac{f}{{(A l^{3} a n u m b e^{2} (r^{1} r^{1}) s)}^{2}}$

Step 10.1.14

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

$- \frac{f}{{(A l^{3} a n u m b e^{2} r^{1 + 1} s)}^{2}}$

Step 10.1.15

Add $1$ and $1$ .

$- \frac{f}{{(A l^{3} a n u m b e^{2} r^{2} s)}^{2}}$

Step 10.2

Apply the product rule to $A l^{3} a n u m b e^{2} r^{2} s$ .

$- \frac{f}{{(A l^{3} a n u m b e^{2} r^{2})}^{2} s^{2}}$

Step 10.3

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 10.3.1

Apply the product rule to $A l^{3} a n u m b e^{2} r^{2}$ .

$- \frac{f}{{(A l^{3} a n u m b e^{2})}^{2} {(r^{2})}^{2} s^{2}}$

Step 10.3.2

Apply the product rule to $A l^{3} a n u m b e^{2}$ .

$- \frac{f}{{(A l^{3} a n u m b)}^{2} {(e^{2})}^{2} {(r^{2})}^{2} s^{2}}$

Step 10.3.3

Apply the product rule to $A l^{3} a n u m b$ .

$- \frac{f}{{(A l^{3} a n u m)}^{2} b^{2} {(e^{2})}^{2} {(r^{2})}^{2} s^{2}}$

Step 10.3.4

Apply the product rule to $A l^{3} a n u m$ .

$- \frac{f}{{(A l^{3} a n u)}^{2} m^{2} b^{2} {(e^{2})}^{2} {(r^{2})}^{2} s^{2}}$

Step 10.3.5

Apply the product rule to $A l^{3} a n u$ .

$- \frac{f}{{(A l^{3} a n)}^{2} u^{2} m^{2} b^{2} {(e^{2})}^{2} {(r^{2})}^{2} s^{2}}$

Step 10.3.6

Apply the product rule to $A l^{3} a n$ .

$- \frac{f}{{(A l^{3} a)}^{2} n^{2} u^{2} m^{2} b^{2} {(e^{2})}^{2} {(r^{2})}^{2} s^{2}}$

Step 10.3.7

Apply the product rule to $A l^{3} a$ .

$- \frac{f}{{(A l^{3})}^{2} a^{2} n^{2} u^{2} m^{2} b^{2} {(e^{2})}^{2} {(r^{2})}^{2} s^{2}}$

Step 10.3.8

Apply the product rule to $A l^{3}$ .

$- \frac{f}{A^{2} {(l^{3})}^{2} a^{2} n^{2} u^{2} m^{2} b^{2} {(e^{2})}^{2} {(r^{2})}^{2} s^{2}}$

Step 10.4

Multiply the exponents in ${(l^{3})}^{2}$ .

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Step 10.4.1

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

$- \frac{f}{A^{2} l^{3 \cdot 2} a^{2} n^{2} u^{2} m^{2} b^{2} {(e^{2})}^{2} {(r^{2})}^{2} s^{2}}$

Step 10.4.2

Multiply $3$ by $2$ .

$- \frac{f}{A^{2} l^{6} a^{2} n^{2} u^{2} m^{2} b^{2} {(e^{2})}^{2} {(r^{2})}^{2} s^{2}}$

Step 10.5

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

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Step 10.5.1

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

$- \frac{f}{A^{2} l^{6} a^{2} n^{2} u^{2} m^{2} b^{2} e^{2 \cdot 2} {(r^{2})}^{2} s^{2}}$

Step 10.5.2

Multiply $2$ by $2$ .

$- \frac{f}{A^{2} l^{6} a^{2} n^{2} u^{2} m^{2} b^{2} e^{4} {(r^{2})}^{2} s^{2}}$

Step 10.6

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

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Step 10.6.1

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

$- \frac{f}{A^{2} l^{6} a^{2} n^{2} u^{2} m^{2} b^{2} e^{4} r^{2 \cdot 2} s^{2}}$

Step 10.6.2

Multiply $2$ by $2$ .

$- \frac{f}{A^{2} l^{6} a^{2} n^{2} u^{2} m^{2} b^{2} e^{4} r^{4} s^{2}}$

Step 11

Since the first derivative test failed, there are no local extrema.

No Local Extrema

Step 12