Calculus Examples

$y = 9 x^{\frac{1}{2}} + x^{\frac{3}{2}}$ ; $x = 9$

Step 1

Find the corresponding $y$ -value to $x = 9$ .

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Step 1.1

Substitute $9$ in for $x$ .

$y = 9 {(9)}^{\frac{1}{2}} + {(9)}^{\frac{3}{2}}$

Step 1.2

Solve for $y$ .

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Step 1.2.1

Remove parentheses.

$y = 9 \cdot 9^{\frac{1}{2}} + {(9)}^{\frac{3}{2}}$

Step 1.2.2

Remove parentheses.

$y = 9 \cdot 9^{\frac{1}{2}} + 9^{\frac{3}{2}}$

Step 1.2.3

Remove parentheses.

$y = 9 {(9)}^{\frac{1}{2}} + {(9)}^{\frac{3}{2}}$

Step 1.2.4

Simplify $9 {(9)}^{\frac{1}{2}} + {(9)}^{\frac{3}{2}}$ .

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Step 1.2.4.1

Simplify each term.

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Step 1.2.4.1.1

Multiply $9$ by ${(9)}^{\frac{1}{2}}$ by adding the exponents.

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Step 1.2.4.1.1.1

Multiply $9$ by ${(9)}^{\frac{1}{2}}$ .

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Step 1.2.4.1.1.1.1

Raise $9$ to the power of $1$ .

$y = 9^{1} {(9)}^{\frac{1}{2}} + {(9)}^{\frac{3}{2}}$

Step 1.2.4.1.1.1.2

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

$y = 9^{1 + \frac{1}{2}} + {(9)}^{\frac{3}{2}}$

Step 1.2.4.1.1.2

Write $1$ as a fraction with a common denominator.

$y = 9^{\frac{2}{2} + \frac{1}{2}} + {(9)}^{\frac{3}{2}}$

Step 1.2.4.1.1.3

Combine the numerators over the common denominator.

$y = 9^{\frac{2 + 1}{2}} + {(9)}^{\frac{3}{2}}$

Step 1.2.4.1.1.4

Add $2$ and $1$ .

$y = 9^{\frac{3}{2}} + {(9)}^{\frac{3}{2}}$

Step 1.2.4.1.2

Rewrite $9$ as $3^{2}$ .

$y = {(3^{2})}^{\frac{3}{2}} + {(9)}^{\frac{3}{2}}$

Step 1.2.4.1.3

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

$y = 3^{2 (\frac{3}{2})} + {(9)}^{\frac{3}{2}}$

Step 1.2.4.1.4

Cancel the common factor of $2$ .

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Step 1.2.4.1.4.1

Cancel the common factor.

$y = 3^{2 (\frac{3}{2})} + {(9)}^{\frac{3}{2}}$

Step 1.2.4.1.4.2

Rewrite the expression.

$y = 3^{3} + {(9)}^{\frac{3}{2}}$

Step 1.2.4.1.5

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

$y = 27 + {(9)}^{\frac{3}{2}}$

Step 1.2.4.1.6

Rewrite $9$ as $3^{2}$ .

$y = 27 + {(3^{2})}^{\frac{3}{2}}$

Step 1.2.4.1.7

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

$y = 27 + 3^{2 (\frac{3}{2})}$

Step 1.2.4.1.8

Cancel the common factor of $2$ .

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Step 1.2.4.1.8.1

Cancel the common factor.

$y = 27 + 3^{2 (\frac{3}{2})}$

Step 1.2.4.1.8.2

Rewrite the expression.

$y = 27 + 3^{3}$

Step 1.2.4.1.9

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

$y = 27 + 27$

Step 1.2.4.2

Add $27$ and $27$ .

$y = 54$

Step 2

Find the first derivative and evaluate at $x = 9$ and $y = 54$ to find the slope of the tangent line.

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Step 2.1

By the Sum Rule, the derivative of $9 x^{\frac{1}{2}} + x^{\frac{3}{2}}$ with respect to $x$ is $\frac{d}{d x} [9 x^{\frac{1}{2}}] + \frac{d}{d x} [x^{\frac{3}{2}}]$ .

$\frac{d}{d x} [9 x^{\frac{1}{2}}] + \frac{d}{d x} [x^{\frac{3}{2}}]$

Step 2.2

Evaluate $\frac{d}{d x} [9 x^{\frac{1}{2}}]$ .

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Step 2.2.1

Since $9$ is constant with respect to $x$ , the derivative of $9 x^{\frac{1}{2}}$ with respect to $x$ is $9 \frac{d}{d x} [x^{\frac{1}{2}}]$ .

$9 \frac{d}{d x} [x^{\frac{1}{2}}] + \frac{d}{d x} [x^{\frac{3}{2}}]$

Step 2.2.2

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

$9 (\frac{1}{2} x^{\frac{1}{2} - 1}) + \frac{d}{d x} [x^{\frac{3}{2}}]$

Step 2.2.3

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

$9 (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}}) + \frac{d}{d x} [x^{\frac{3}{2}}]$

Step 2.2.4

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

$9 (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}}) + \frac{d}{d x} [x^{\frac{3}{2}}]$

Step 2.2.5

Combine the numerators over the common denominator.

$9 (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}}) + \frac{d}{d x} [x^{\frac{3}{2}}]$

Step 2.2.6

Simplify the numerator.

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Step 2.2.6.1

Multiply $- 1$ by $2$ .

$9 (\frac{1}{2} x^{\frac{1 - 2}{2}}) + \frac{d}{d x} [x^{\frac{3}{2}}]$

Step 2.2.6.2

Subtract $2$ from $1$ .

$9 (\frac{1}{2} x^{\frac{- 1}{2}}) + \frac{d}{d x} [x^{\frac{3}{2}}]$

Step 2.2.7

Move the negative in front of the fraction.

$9 (\frac{1}{2} x^{- \frac{1}{2}}) + \frac{d}{d x} [x^{\frac{3}{2}}]$

Step 2.2.8

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

$9 \frac{x^{- \frac{1}{2}}}{2} + \frac{d}{d x} [x^{\frac{3}{2}}]$

Step 2.2.9

Combine $9$ and $\frac{x^{- \frac{1}{2}}}{2}$ .

$\frac{9 x^{- \frac{1}{2}}}{2} + \frac{d}{d x} [x^{\frac{3}{2}}]$

Step 2.2.10

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

$\frac{9}{2 x^{\frac{1}{2}}} + \frac{d}{d x} [x^{\frac{3}{2}}]$

Step 2.3

Evaluate $\frac{d}{d x} [x^{\frac{3}{2}}]$ .

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Step 2.3.1

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

$\frac{9}{2 x^{\frac{1}{2}}} + \frac{3}{2} x^{\frac{3}{2} - 1}$

Step 2.3.2

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

$\frac{9}{2 x^{\frac{1}{2}}} + \frac{3}{2} x^{\frac{3}{2} - 1 \cdot \frac{2}{2}}$

Step 2.3.3

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

$\frac{9}{2 x^{\frac{1}{2}}} + \frac{3}{2} x^{\frac{3}{2} + \frac{- 1 \cdot 2}{2}}$

Step 2.3.4

Combine the numerators over the common denominator.

$\frac{9}{2 x^{\frac{1}{2}}} + \frac{3}{2} x^{\frac{3 - 1 \cdot 2}{2}}$

Step 2.3.5

Simplify the numerator.

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Step 2.3.5.1

Multiply $- 1$ by $2$ .

$\frac{9}{2 x^{\frac{1}{2}}} + \frac{3}{2} x^{\frac{3 - 2}{2}}$

Step 2.3.5.2

Subtract $2$ from $3$ .

$\frac{9}{2 x^{\frac{1}{2}}} + \frac{3}{2} x^{\frac{1}{2}}$

Step 2.4

Simplify.

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Step 2.4.1

Reorder terms.

$\frac{3}{2} x^{\frac{1}{2}} + \frac{9}{2 x^{\frac{1}{2}}}$

Step 2.4.2

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

$\frac{3 x^{\frac{1}{2}}}{2} + \frac{9}{2 x^{\frac{1}{2}}}$

Step 2.5

Evaluate the derivative at $x = 9$ .

$\frac{3 {(9)}^{\frac{1}{2}}}{2} + \frac{9}{2 {(9)}^{\frac{1}{2}}}$

Step 2.6

Simplify.

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Step 2.6.1

Simplify each term.

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Step 2.6.1.1

Simplify the numerator.

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Step 2.6.1.1.1

Rewrite $9$ as $3^{2}$ .

$\frac{3 \cdot {(3^{2})}^{\frac{1}{2}}}{2} + \frac{9}{2 {(9)}^{\frac{1}{2}}}$

Step 2.6.1.1.2

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

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Step 2.6.1.1.2.1

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

$\frac{3 \cdot 3^{2 (\frac{1}{2})}}{2} + \frac{9}{2 {(9)}^{\frac{1}{2}}}$

Step 2.6.1.1.2.2

Cancel the common factor of $2$ .

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Step 2.6.1.1.2.2.1

Cancel the common factor.

$\frac{3 \cdot 3^{2 (\frac{1}{2})}}{2} + \frac{9}{2 {(9)}^{\frac{1}{2}}}$

Step 2.6.1.1.2.2.2

Rewrite the expression.

$\frac{3 \cdot 3^{1}}{2} + \frac{9}{2 {(9)}^{\frac{1}{2}}}$

Step 2.6.1.1.3

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

$\frac{3^{1 + 1}}{2} + \frac{9}{2 {(9)}^{\frac{1}{2}}}$

Step 2.6.1.1.4

Add $1$ and $1$ .

$\frac{3^{2}}{2} + \frac{9}{2 {(9)}^{\frac{1}{2}}}$

Step 2.6.1.2

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

$\frac{9}{2} + \frac{9}{2 {(9)}^{\frac{1}{2}}}$

Step 2.6.1.3

Move ${(9)}^{\frac{1}{2}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$\frac{9}{2} + \frac{9 \cdot 9^{- \frac{1}{2}}}{2}$

Step 2.6.1.4

Multiply $9$ by $9^{- \frac{1}{2}}$ by adding the exponents.

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Step 2.6.1.4.1

Multiply $9$ by $9^{- \frac{1}{2}}$ .

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Step 2.6.1.4.1.1

Raise $9$ to the power of $1$ .

$\frac{9}{2} + \frac{9^{1} \cdot 9^{- \frac{1}{2}}}{2}$

Step 2.6.1.4.1.2

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

$\frac{9}{2} + \frac{9^{1 - \frac{1}{2}}}{2}$

Step 2.6.1.4.2

Write $1$ as a fraction with a common denominator.

$\frac{9}{2} + \frac{9^{\frac{2}{2} - \frac{1}{2}}}{2}$

Step 2.6.1.4.3

Combine the numerators over the common denominator.

$\frac{9}{2} + \frac{9^{\frac{2 - 1}{2}}}{2}$

Step 2.6.1.4.4

Subtract $1$ from $2$ .

$\frac{9}{2} + \frac{9^{\frac{1}{2}}}{2}$

Step 2.6.1.5

Simplify the numerator.

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Step 2.6.1.5.1

Rewrite $9$ as $3^{2}$ .

$\frac{9}{2} + \frac{{(3^{2})}^{\frac{1}{2}}}{2}$

Step 2.6.1.5.2

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

$\frac{9}{2} + \frac{3^{2 (\frac{1}{2})}}{2}$

Step 2.6.1.5.3

Cancel the common factor of $2$ .

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Step 2.6.1.5.3.1

Cancel the common factor.

$\frac{9}{2} + \frac{3^{2 (\frac{1}{2})}}{2}$

Step 2.6.1.5.3.2

Rewrite the expression.

$\frac{9}{2} + \frac{3^{1}}{2}$

Step 2.6.1.5.4

Evaluate the exponent.

$\frac{9}{2} + \frac{3}{2}$

Step 2.6.2

Combine fractions.

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Step 2.6.2.1

Combine the numerators over the common denominator.

$\frac{9 + 3}{2}$

Step 2.6.2.2

Simplify the expression.

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Step 2.6.2.2.1

Add $9$ and $3$ .

$\frac{12}{2}$

Step 2.6.2.2.2

Divide $12$ by $2$ .

$6$

Step 3

Plug the slope and point values into the point-slope formula and solve for $y$ .

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Step 3.1

Use the slope $6$ and a given point $(9, 54)$ to substitute for $x_{1}$ and $y_{1}$ in the point-slope form $y - y_{1} = m (x - x_{1})$ , which is derived from the slope equation $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (54) = 6 \cdot (x - (9))$

Step 3.2

Simplify the equation and keep it in point-slope form.

$y - 54 = 6 \cdot (x - 9)$

Step 3.3

Solve for $y$ .

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Step 3.3.1

Simplify $6 \cdot (x - 9)$ .

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Step 3.3.1.1

Rewrite.

$y - 54 = 0 + 0 + 6 \cdot (x - 9)$

Step 3.3.1.2

Simplify by adding zeros.

$y - 54 = 6 \cdot (x - 9)$

Step 3.3.1.3

Apply the distributive property.

$y - 54 = 6 x + 6 \cdot - 9$

Step 3.3.1.4

Multiply $6$ by $- 9$ .

$y - 54 = 6 x - 54$

Step 3.3.2

Move all terms not containing $y$ to the right side of the equation.

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Step 3.3.2.1

Add $54$ to both sides of the equation.

$y = 6 x - 54 + 54$

Step 3.3.2.2

Combine the opposite terms in $6 x - 54 + 54$ .

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Step 3.3.2.2.1

Add $- 54$ and $54$ .

$y = 6 x + 0$

Step 3.3.2.2.2

Add $6 x$ and $0$ .

$y = 6 x$

Step 4