Calculus Examples

$y = \frac{7}{\sqrt{x}}$

Step 1

Write $y = \frac{7}{\sqrt{x}}$ as a function.

$f (x) = \frac{7}{\sqrt{x}}$

Step 2

Find the first derivative of the function.

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$\frac{d}{d x} [\frac{7}{x^{\frac{1}{2}}}]$

Step 2.2

Since $7$ is constant with respect to $x$ , the derivative of $\frac{7}{x^{\frac{1}{2}}}$ with respect to $x$ is $7 \frac{d}{d x} [\frac{1}{x^{\frac{1}{2}}}]$ .

$7 \frac{d}{d x} [\frac{1}{x^{\frac{1}{2}}}]$

Step 2.3

Apply basic rules of exponents.

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Step 2.3.1

Rewrite $\frac{1}{x^{\frac{1}{2}}}$ as ${(x^{\frac{1}{2}})}^{- 1}$ .

$7 \frac{d}{d x} [{(x^{\frac{1}{2}})}^{- 1}]$

Step 2.3.2

Multiply the exponents in ${(x^{\frac{1}{2}})}^{- 1}$ .

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Step 2.3.2.1

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

$7 \frac{d}{d x} [x^{\frac{1}{2} \cdot - 1}]$

Step 2.3.2.2

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

$7 \frac{d}{d x} [x^{\frac{- 1}{2}}]$

Step 2.3.2.3

Move the negative in front of the fraction.

$7 \frac{d}{d x} [x^{- \frac{1}{2}}]$

Step 2.4

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

$7 (- \frac{1}{2} x^{- \frac{1}{2} - 1})$

Step 2.5

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$7 (- \frac{1}{2} x^{- \frac{1}{2} - 1 \cdot \frac{2}{2}})$

Step 2.6

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

$7 (- \frac{1}{2} x^{- \frac{1}{2} + \frac{- 1 \cdot 2}{2}})$

Step 2.7

Combine the numerators over the common denominator.

$7 (- \frac{1}{2} x^{\frac{- 1 - 1 \cdot 2}{2}})$

Step 2.8

Simplify the numerator.

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Step 2.8.1

Multiply $- 1$ by $2$ .

$7 (- \frac{1}{2} x^{\frac{- 1 - 2}{2}})$

Step 2.8.2

Subtract $2$ from $- 1$ .

$7 (- \frac{1}{2} x^{\frac{- 3}{2}})$

Step 2.9

Move the negative in front of the fraction.

$7 (- \frac{1}{2} x^{- \frac{3}{2}})$

Step 2.10

Combine $x^{- \frac{3}{2}}$ and $\frac{1}{2}$ .

$7 (- \frac{x^{- \frac{3}{2}}}{2})$

Step 2.11

Multiply $- 1$ by $7$ .

$- 7 \frac{x^{- \frac{3}{2}}}{2}$

Step 2.12

Combine $- 7$ and $\frac{x^{- \frac{3}{2}}}{2}$ .

$\frac{- 7 x^{- \frac{3}{2}}}{2}$

Step 2.13

Simplify the expression.

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Step 2.13.1

Move $x^{- \frac{3}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{- 7}{2 x^{\frac{3}{2}}}$

Step 2.13.2

Move the negative in front of the fraction.

$- \frac{7}{2 x^{\frac{3}{2}}}$

Step 3

Find the second derivative of the function.

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Step 3.1

Since $- \frac{7}{2}$ is constant with respect to $x$ , the derivative of $- \frac{7}{2 x^{\frac{3}{2}}}$ with respect to $x$ is $- \frac{7}{2} \frac{d}{d x} [\frac{1}{x^{\frac{3}{2}}}]$ .

$f'' (x) = - \frac{7}{2} \cdot \frac{d}{d x} \frac{1}{x^{\frac{3}{2}}}$

Step 3.2

Apply basic rules of exponents.

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Step 3.2.1

Rewrite $\frac{1}{x^{\frac{3}{2}}}$ as ${(x^{\frac{3}{2}})}^{- 1}$ .

$f'' (x) = - \frac{7}{2} \cdot \frac{d}{d x} {(x^{\frac{3}{2}})}^{- 1}$

Step 3.2.2

Multiply the exponents in ${(x^{\frac{3}{2}})}^{- 1}$ .

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Step 3.2.2.1

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

$f'' (x) = - \frac{7}{2} \cdot \frac{d}{d x} x^{\frac{3}{2} \cdot - 1}$

Step 3.2.2.2

Multiply $\frac{3}{2} \cdot - 1$ .

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Step 3.2.2.2.1

Combine $\frac{3}{2}$ and $- 1$ .

$f'' (x) = - \frac{7}{2} \cdot \frac{d}{d x} x^{\frac{3 \cdot - 1}{2}}$

Step 3.2.2.2.2

Multiply $3$ by $- 1$ .

$f'' (x) = - \frac{7}{2} \cdot \frac{d}{d x} x^{\frac{- 3}{2}}$

Step 3.2.2.3

Move the negative in front of the fraction.

$f'' (x) = - \frac{7}{2} \cdot \frac{d}{d x} x^{- \frac{3}{2}}$

Step 3.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - \frac{3}{2}$ .

$f'' (x) = - \frac{7}{2} \cdot (- \frac{3}{2} x^{- \frac{3}{2} - 1})$

Step 3.4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f'' (x) = - \frac{7}{2} \cdot (- \frac{3}{2} x^{- \frac{3}{2} - 1 \cdot \frac{2}{2}})$

Step 3.5

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

$f'' (x) = - \frac{7}{2} \cdot (- \frac{3}{2} x^{- \frac{3}{2} + \frac{- 1 \cdot 2}{2}})$

Step 3.6

Combine the numerators over the common denominator.

$f'' (x) = - \frac{7}{2} \cdot (- \frac{3}{2} x^{\frac{- 3 - 1 \cdot 2}{2}})$

Step 3.7

Simplify the numerator.

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Step 3.7.1

Multiply $- 1$ by $2$ .

$f'' (x) = - \frac{7}{2} \cdot (- \frac{3}{2} x^{\frac{- 3 - 2}{2}})$

Step 3.7.2

Subtract $2$ from $- 3$ .

$f'' (x) = - \frac{7}{2} \cdot (- \frac{3}{2} x^{\frac{- 5}{2}})$

Step 3.8

Move the negative in front of the fraction.

$f'' (x) = - \frac{7}{2} \cdot (- \frac{3}{2} x^{- \frac{5}{2}})$

Step 3.9

Combine $x^{- \frac{5}{2}}$ and $\frac{3}{2}$ .

$f'' (x) = - \frac{7}{2} \cdot (- \frac{x^{- \frac{5}{2}} \cdot 3}{2})$

Step 3.10

Multiply.

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Step 3.10.1

Multiply $- 1$ by $- 1$ .

$f'' (x) = 1 (\frac{7}{2}) \cdot \frac{x^{- \frac{5}{2}} \cdot 3}{2}$

Step 3.10.2

Multiply $\frac{7}{2}$ by $1$ .

$f'' (x) = \frac{7}{2} \cdot \frac{x^{- \frac{5}{2}} \cdot 3}{2}$

Step 3.11

Multiply $\frac{7}{2}$ by $\frac{x^{- \frac{5}{2}} \cdot 3}{2}$ .

$f'' (x) = \frac{7 (x^{- \frac{5}{2}} \cdot 3)}{2 \cdot 2}$

Step 3.12

Multiply.

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Step 3.12.1

Multiply $3$ by $7$ .

$f'' (x) = \frac{21 x^{- \frac{5}{2}}}{2 \cdot 2}$

Step 3.12.2

Multiply $2$ by $2$ .

$f'' (x) = \frac{21 x^{- \frac{5}{2}}}{4}$

Step 3.12.3

Move $x^{- \frac{5}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f'' (x) = \frac{21}{4 x^{\frac{5}{2}}}$

Step 4

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

$- \frac{7}{2 x^{\frac{3}{2}}} = 0$

Step 5

Since there is no value of $x$ that makes the first derivative equal to $0$ , there are no local extrema.

No Local Extrema

Step 6

No Local Extrema

Step 7