Calculus Examples

$y = 5 x^{\frac{1}{2}} + x^{\frac{3}{2}}$ ; $x = 25$

Step 1

Find the corresponding $y$ -value to $x = 25$ .

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Step 1.1

Substitute $25$ in for $x$ .

$y = 5 {(25)}^{\frac{1}{2}} + {(25)}^{\frac{3}{2}}$

Step 1.2

Solve for $y$ .

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Step 1.2.1

Remove parentheses.

$y = 5 \cdot 25^{\frac{1}{2}} + {(25)}^{\frac{3}{2}}$

Step 1.2.2

Remove parentheses.

$y = 5 \cdot 25^{\frac{1}{2}} + 25^{\frac{3}{2}}$

Step 1.2.3

Remove parentheses.

$y = 5 {(25)}^{\frac{1}{2}} + {(25)}^{\frac{3}{2}}$

Step 1.2.4

Simplify $5 {(25)}^{\frac{1}{2}} + {(25)}^{\frac{3}{2}}$ .

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Step 1.2.4.1

Simplify each term.

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Step 1.2.4.1.1

Rewrite $25$ as $5^{2}$ .

$y = 5 {(5^{2})}^{\frac{1}{2}} + {(25)}^{\frac{3}{2}}$

Step 1.2.4.1.2

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

$y = 5 \cdot 5^{2 (\frac{1}{2})} + {(25)}^{\frac{3}{2}}$

Step 1.2.4.1.3

Cancel the common factor of $2$ .

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Step 1.2.4.1.3.1

Cancel the common factor.

$y = 5 \cdot 5^{2 (\frac{1}{2})} + {(25)}^{\frac{3}{2}}$

Step 1.2.4.1.3.2

Rewrite the expression.

$y = 5 \cdot 5^{1} + {(25)}^{\frac{3}{2}}$

Step 1.2.4.1.4

Evaluate the exponent.

$y = 5 \cdot 5 + {(25)}^{\frac{3}{2}}$

Step 1.2.4.1.5

Multiply $5$ by $5$ .

$y = 25 + {(25)}^{\frac{3}{2}}$

Step 1.2.4.1.6

Rewrite $25$ as $5^{2}$ .

$y = 25 + {(5^{2})}^{\frac{3}{2}}$

Step 1.2.4.1.7

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

$y = 25 + 5^{2 (\frac{3}{2})}$

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.

$y = 25 + 5^{2 (\frac{3}{2})}$

Step 1.2.4.1.8.2

Rewrite the expression.

$y = 25 + 5^{3}$

Step 1.2.4.1.9

Raise $5$ to the power of $3$ .

$y = 25 + 125$

Step 1.2.4.2

Add $25$ and $125$ .

$y = 150$

Step 2

Find the first derivative and evaluate at $x = 25$ and $y = 150$ to find the slope of the tangent line.

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Step 2.1

By the Sum Rule, the derivative of $5 x^{\frac{1}{2}} + x^{\frac{3}{2}}$ with respect to $x$ is $\frac{d}{d x} [5 x^{\frac{1}{2}}] + \frac{d}{d x} [x^{\frac{3}{2}}]$ .

$\frac{d}{d x} [5 x^{\frac{1}{2}}] + \frac{d}{d x} [x^{\frac{3}{2}}]$

Step 2.2

Evaluate $\frac{d}{d x} [5 x^{\frac{1}{2}}]$ .

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Step 2.2.1

Since $5$ is constant with respect to $x$ , the derivative of $5 x^{\frac{1}{2}}$ with respect to $x$ is $5 \frac{d}{d x} [x^{\frac{1}{2}}]$ .

$5 \frac{d}{d x} [x^{\frac{1}{2}}] + \frac{d}{d x} [x^{\frac{3}{2}}]$

Step 2.2.2

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

$5 (\frac{1}{2} x^{\frac{1}{2} - 1}) + \frac{d}{d x} [x^{\frac{3}{2}}]$

Step 2.2.3

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

$5 (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}}) + \frac{d}{d x} [x^{\frac{3}{2}}]$

Step 2.2.4

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

$5 (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}}) + \frac{d}{d x} [x^{\frac{3}{2}}]$

Step 2.2.5

Combine the numerators over the common denominator.

$5 (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}}) + \frac{d}{d x} [x^{\frac{3}{2}}]$

Step 2.2.6

Simplify the numerator.

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Step 2.2.6.1

Multiply $- 1$ by $2$ .

$5 (\frac{1}{2} x^{\frac{1 - 2}{2}}) + \frac{d}{d x} [x^{\frac{3}{2}}]$

Step 2.2.6.2

Subtract $2$ from $1$ .

$5 (\frac{1}{2} x^{\frac{- 1}{2}}) + \frac{d}{d x} [x^{\frac{3}{2}}]$

Step 2.2.7

Move the negative in front of the fraction.

$5 (\frac{1}{2} x^{- \frac{1}{2}}) + \frac{d}{d x} [x^{\frac{3}{2}}]$

Step 2.2.8

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

$5 \frac{x^{- \frac{1}{2}}}{2} + \frac{d}{d x} [x^{\frac{3}{2}}]$

Step 2.2.9

Combine $5$ and $\frac{x^{- \frac{1}{2}}}{2}$ .

$\frac{5 x^{- \frac{1}{2}}}{2} + \frac{d}{d x} [x^{\frac{3}{2}}]$

Step 2.2.10

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

$\frac{5}{2 x^{\frac{1}{2}}} + \frac{d}{d x} [x^{\frac{3}{2}}]$

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 $n = \frac{3}{2}$ .

$\frac{5}{2 x^{\frac{1}{2}}} + \frac{3}{2} x^{\frac{3}{2} - 1}$

Step 2.3.2

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

$\frac{5}{2 x^{\frac{1}{2}}} + \frac{3}{2} x^{\frac{3}{2} - 1 \cdot \frac{2}{2}}$

Step 2.3.3

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

$\frac{5}{2 x^{\frac{1}{2}}} + \frac{3}{2} x^{\frac{3}{2} + \frac{- 1 \cdot 2}{2}}$

Step 2.3.4

Combine the numerators over the common denominator.

$\frac{5}{2 x^{\frac{1}{2}}} + \frac{3}{2} x^{\frac{3 - 1 \cdot 2}{2}}$

Step 2.3.5

Simplify the numerator.

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Step 2.3.5.1

Multiply $- 1$ by $2$ .

$\frac{5}{2 x^{\frac{1}{2}}} + \frac{3}{2} x^{\frac{3 - 2}{2}}$

Step 2.3.5.2

Subtract $2$ from $3$ .

$\frac{5}{2 x^{\frac{1}{2}}} + \frac{3}{2} x^{\frac{1}{2}}$

Step 2.4

Simplify.

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Step 2.4.1

Reorder terms.

$\frac{3}{2} x^{\frac{1}{2}} + \frac{5}{2 x^{\frac{1}{2}}}$

Step 2.4.2

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

$\frac{3 x^{\frac{1}{2}}}{2} + \frac{5}{2 x^{\frac{1}{2}}}$

Step 2.5

Evaluate the derivative at $x = 25$ .

$\frac{3 {(25)}^{\frac{1}{2}}}{2} + \frac{5}{2 {(25)}^{\frac{1}{2}}}$

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 $25$ as $5^{2}$ .

$\frac{3 \cdot {(5^{2})}^{\frac{1}{2}}}{2} + \frac{5}{2 {(25)}^{\frac{1}{2}}}$

Step 2.6.1.1.2

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

$\frac{3 \cdot 5^{2 (\frac{1}{2})}}{2} + \frac{5}{2 {(25)}^{\frac{1}{2}}}$

Step 2.6.1.1.3

Cancel the common factor of $2$ .

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Step 2.6.1.1.3.1

Cancel the common factor.

$\frac{3 \cdot 5^{2 (\frac{1}{2})}}{2} + \frac{5}{2 {(25)}^{\frac{1}{2}}}$

Step 2.6.1.1.3.2

Rewrite the expression.

$\frac{3 \cdot 5^{1}}{2} + \frac{5}{2 {(25)}^{\frac{1}{2}}}$

Step 2.6.1.1.4

Evaluate the exponent.

$\frac{3 \cdot 5}{2} + \frac{5}{2 {(25)}^{\frac{1}{2}}}$

Step 2.6.1.2

Multiply $3$ by $5$ .

$\frac{15}{2} + \frac{5}{2 {(25)}^{\frac{1}{2}}}$

Step 2.6.1.3

Simplify the denominator.

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Step 2.6.1.3.1

Rewrite $25$ as $5^{2}$ .

$\frac{15}{2} + \frac{5}{2 \cdot {(5^{2})}^{\frac{1}{2}}}$

Step 2.6.1.3.2

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

$\frac{15}{2} + \frac{5}{2 \cdot 5^{2 (\frac{1}{2})}}$

Step 2.6.1.3.3

Cancel the common factor of $2$ .

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Step 2.6.1.3.3.1

Cancel the common factor.

$\frac{15}{2} + \frac{5}{2 \cdot 5^{2 (\frac{1}{2})}}$

Step 2.6.1.3.3.2

Rewrite the expression.

$\frac{15}{2} + \frac{5}{2 \cdot 5^{1}}$

Step 2.6.1.3.4

Evaluate the exponent.

$\frac{15}{2} + \frac{5}{2 \cdot 5}$

Step 2.6.1.4

Multiply $2$ by $5$ .

$\frac{15}{2} + \frac{5}{10}$

Step 2.6.1.5

Cancel the common factor of $5$ and $10$ .

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Step 2.6.1.5.1

Factor $5$ out of $5$ .

$\frac{15}{2} + \frac{5 (1)}{10}$

Step 2.6.1.5.2

Cancel the common factors.

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Step 2.6.1.5.2.1

Factor $5$ out of $10$ .

$\frac{15}{2} + \frac{5 \cdot 1}{5 \cdot 2}$

Step 2.6.1.5.2.2

Cancel the common factor.

$\frac{15}{2} + \frac{5 \cdot 1}{5 \cdot 2}$

Step 2.6.1.5.2.3

Rewrite the expression.

$\frac{15}{2} + \frac{1}{2}$

Step 2.6.2

Combine fractions.

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Step 2.6.2.1

Combine the numerators over the common denominator.

$\frac{15 + 1}{2}$

Step 2.6.2.2

Simplify the expression.

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Step 2.6.2.2.1

Add $15$ and $1$ .

$\frac{16}{2}$

Step 2.6.2.2.2

Divide $16$ by $2$ .

$8$

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 $8$ and a given point $(25, 150)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (150) = 8 \cdot (x - (25))$

Step 3.2

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

$y - 150 = 8 \cdot (x - 25)$

Step 3.3

Solve for $y$ .

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Step 3.3.1

Simplify $8 \cdot (x - 25)$ .

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Step 3.3.1.1

Rewrite.

$y - 150 = 0 + 0 + 8 \cdot (x - 25)$

Step 3.3.1.2

Simplify by adding zeros.

$y - 150 = 8 \cdot (x - 25)$

Step 3.3.1.3

Apply the distributive property.

$y - 150 = 8 x + 8 \cdot - 25$

Step 3.3.1.4

Multiply $8$ by $- 25$ .

$y - 150 = 8 x - 200$

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 $150$ to both sides of the equation.

$y = 8 x - 200 + 150$

Step 3.3.2.2

Add $- 200$ and $150$ .

$y = 8 x - 50$

Step 4