Calculus Examples

$f (x) = - 4 x^{\frac{1}{2}} - 5 x^{- 1} - 5$ at the point $(4, - \frac{57}{4})$

Step 1

Find the first derivative and evaluate at $x = 4$ and $y = - \frac{57}{4}$ to find the slope of the tangent line.

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Step 1.1

By the Sum Rule, the derivative of $- 4 x^{\frac{1}{2}} - 5 x^{- 1} - 5$ with respect to $x$ is $\frac{d}{d x} [- 4 x^{\frac{1}{2}}] + \frac{d}{d x} [- 5 x^{- 1}] + \frac{d}{d x} [- 5]$ .

$\frac{d}{d x} [- 4 x^{\frac{1}{2}}] + \frac{d}{d x} [- 5 x^{- 1}] + \frac{d}{d x} [- 5]$

Step 1.2

Evaluate $\frac{d}{d x} [- 4 x^{\frac{1}{2}}]$ .

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Step 1.2.1

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4 x^{\frac{1}{2}}$ with respect to $x$ is $- 4 \frac{d}{d x} [x^{\frac{1}{2}}]$ .

$- 4 \frac{d}{d x} [x^{\frac{1}{2}}] + \frac{d}{d x} [- 5 x^{- 1}] + \frac{d}{d x} [- 5]$

Step 1.2.2

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

$- 4 (\frac{1}{2} x^{\frac{1}{2} - 1}) + \frac{d}{d x} [- 5 x^{- 1}] + \frac{d}{d x} [- 5]$

Step 1.2.3

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

$- 4 (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}}) + \frac{d}{d x} [- 5 x^{- 1}] + \frac{d}{d x} [- 5]$

Step 1.2.4

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

$- 4 (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}}) + \frac{d}{d x} [- 5 x^{- 1}] + \frac{d}{d x} [- 5]$

Step 1.2.5

Combine the numerators over the common denominator.

$- 4 (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}}) + \frac{d}{d x} [- 5 x^{- 1}] + \frac{d}{d x} [- 5]$

Step 1.2.6

Simplify the numerator.

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Step 1.2.6.1

Multiply $- 1$ by $2$ .

$- 4 (\frac{1}{2} x^{\frac{1 - 2}{2}}) + \frac{d}{d x} [- 5 x^{- 1}] + \frac{d}{d x} [- 5]$

Step 1.2.6.2

Subtract $2$ from $1$ .

$- 4 (\frac{1}{2} x^{\frac{- 1}{2}}) + \frac{d}{d x} [- 5 x^{- 1}] + \frac{d}{d x} [- 5]$

Step 1.2.7

Move the negative in front of the fraction.

$- 4 (\frac{1}{2} x^{- \frac{1}{2}}) + \frac{d}{d x} [- 5 x^{- 1}] + \frac{d}{d x} [- 5]$

Step 1.2.8

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

$- 4 \frac{x^{- \frac{1}{2}}}{2} + \frac{d}{d x} [- 5 x^{- 1}] + \frac{d}{d x} [- 5]$

Step 1.2.9

Combine $- 4$ and $\frac{x^{- \frac{1}{2}}}{2}$ .

$\frac{- 4 x^{- \frac{1}{2}}}{2} + \frac{d}{d x} [- 5 x^{- 1}] + \frac{d}{d x} [- 5]$

Step 1.2.10

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

$\frac{- 4}{2 x^{\frac{1}{2}}} + \frac{d}{d x} [- 5 x^{- 1}] + \frac{d}{d x} [- 5]$

Step 1.2.11

Factor $2$ out of $- 4$ .

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

Step 1.2.12

Cancel the common factors.

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Step 1.2.12.1

Factor $2$ out of $2 x^{\frac{1}{2}}$ .

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

Step 1.2.12.2

Cancel the common factor.

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

Step 1.2.12.3

Rewrite the expression.

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

Step 1.2.13

Move the negative in front of the fraction.

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

Step 1.3

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

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Step 1.3.1

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

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

Step 1.3.2

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

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

Step 1.3.3

Multiply $- 1$ by $- 5$ .

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

Step 1.4

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5$ with respect to $x$ is $0$ .

$- \frac{2}{x^{\frac{1}{2}}} + 5 x^{- 2} + 0$

Step 1.5

Simplify.

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Step 1.5.1

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

$- \frac{2}{x^{\frac{1}{2}}} + 5 \frac{1}{x^{2}} + 0$

Step 1.5.2

Combine terms.

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Step 1.5.2.1

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

$- \frac{2}{x^{\frac{1}{2}}} + \frac{5}{x^{2}} + 0$

Step 1.5.2.2

Add $- \frac{2}{x^{\frac{1}{2}}} + \frac{5}{x^{2}}$ and $0$ .

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

Step 1.5.3

Reorder terms.

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

Step 1.6

Evaluate the derivative at $x = 4$ .

$\frac{5}{{(4)}^{2}} - \frac{2}{{(4)}^{\frac{1}{2}}}$

Step 1.7

Simplify.

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Step 1.7.1

Simplify each term.

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Step 1.7.1.1

Raise $4$ to the power of $2$ .

$\frac{5}{16} - \frac{2}{{(4)}^{\frac{1}{2}}}$

Step 1.7.1.2

Simplify the denominator.

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Step 1.7.1.2.1

Rewrite $4$ as $2^{2}$ .

$\frac{5}{16} - \frac{2}{{(2^{2})}^{\frac{1}{2}}}$

Step 1.7.1.2.2

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

$\frac{5}{16} - \frac{2}{2^{2 (\frac{1}{2})}}$

Step 1.7.1.2.3

Cancel the common factor of $2$ .

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Step 1.7.1.2.3.1

Cancel the common factor.

$\frac{5}{16} - \frac{2}{2^{2 (\frac{1}{2})}}$

Step 1.7.1.2.3.2

Rewrite the expression.

$\frac{5}{16} - \frac{2}{2^{1}}$

Step 1.7.1.2.4

Evaluate the exponent.

$\frac{5}{16} - \frac{2}{2}$

Step 1.7.1.3

Cancel the common factor of $2$ .

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Step 1.7.1.3.1

Cancel the common factor.

$\frac{5}{16} - \frac{2}{2}$

Step 1.7.1.3.2

Rewrite the expression.

$\frac{5}{16} - 1 \cdot 1$

Step 1.7.1.4

Multiply $- 1$ by $1$ .

$\frac{5}{16} - 1$

Step 1.7.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{16}{16}$ .

$\frac{5}{16} - 1 \cdot \frac{16}{16}$

Step 1.7.3

Combine $- 1$ and $\frac{16}{16}$ .

$\frac{5}{16} + \frac{- 1 \cdot 16}{16}$

Step 1.7.4

Combine the numerators over the common denominator.

$\frac{5 - 1 \cdot 16}{16}$

Step 1.7.5

Simplify the numerator.

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Step 1.7.5.1

Multiply $- 1$ by $16$ .

$\frac{5 - 16}{16}$

Step 1.7.5.2

Subtract $16$ from $5$ .

$\frac{- 11}{16}$

Step 1.7.6

Move the negative in front of the fraction.

$- \frac{11}{16}$

Step 2

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

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Step 2.1

Use the slope $- \frac{11}{16}$ and a given point $(4, - \frac{57}{4})$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (- \frac{57}{4}) = - \frac{11}{16} \cdot (x - (4))$

Step 2.2

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

$y + \frac{57}{4} = - \frac{11}{16} \cdot (x - 4)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $- \frac{11}{16} \cdot (x - 4)$ .

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Step 2.3.1.1

Rewrite.

$y + \frac{57}{4} = 0 + 0 - \frac{11}{16} \cdot (x - 4)$

Step 2.3.1.2

Simplify terms.

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Step 2.3.1.2.1

Apply the distributive property.

$y + \frac{57}{4} = - \frac{11}{16} x - \frac{11}{16} \cdot - 4$

Step 2.3.1.2.2

Combine $x$ and $\frac{11}{16}$ .

$y + \frac{57}{4} = - \frac{x \cdot 11}{16} - \frac{11}{16} \cdot - 4$

Step 2.3.1.2.3

Cancel the common factor of $4$ .

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Step 2.3.1.2.3.1

Move the leading negative in $- \frac{11}{16}$ into the numerator.

$y + \frac{57}{4} = - \frac{x \cdot 11}{16} + \frac{- 11}{16} \cdot - 4$

Step 2.3.1.2.3.2

Factor $4$ out of $16$ .

$y + \frac{57}{4} = - \frac{x \cdot 11}{16} + \frac{- 11}{4 (4)} \cdot - 4$

Step 2.3.1.2.3.3

Factor $4$ out of $- 4$ .

$y + \frac{57}{4} = - \frac{x \cdot 11}{16} + \frac{- 11}{4 \cdot 4} \cdot (4 \cdot - 1)$

Step 2.3.1.2.3.4

Cancel the common factor.

$y + \frac{57}{4} = - \frac{x \cdot 11}{16} + \frac{- 11}{4 \cdot 4} \cdot (4 \cdot - 1)$

Step 2.3.1.2.3.5

Rewrite the expression.

$y + \frac{57}{4} = - \frac{x \cdot 11}{16} + \frac{- 11}{4} \cdot - 1$

Step 2.3.1.2.4

Combine $\frac{- 11}{4}$ and $- 1$ .

$y + \frac{57}{4} = - \frac{x \cdot 11}{16} + \frac{- 11 \cdot - 1}{4}$

Step 2.3.1.2.5

Multiply $- 11$ by $- 1$ .

$y + \frac{57}{4} = - \frac{x \cdot 11}{16} + \frac{11}{4}$

Step 2.3.1.3

Move $11$ to the left of $x$ .

$y + \frac{57}{4} = - \frac{11 x}{16} + \frac{11}{4}$

Step 2.3.2

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

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Step 2.3.2.1

Subtract $\frac{57}{4}$ from both sides of the equation.

$y = - \frac{11 x}{16} + \frac{11}{4} - \frac{57}{4}$

Step 2.3.2.2

Combine the numerators over the common denominator.

$y = \frac{- 11 x}{16} + \frac{11 - 57}{4}$

Step 2.3.2.3

Subtract $57$ from $11$ .

$y = \frac{- 11 x}{16} + \frac{- 46}{4}$

Step 2.3.2.4

Simplify each term.

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Step 2.3.2.4.1

Move the negative in front of the fraction.

$y = - \frac{11 x}{16} + \frac{- 46}{4}$

Step 2.3.2.4.2

Cancel the common factor of $- 46$ and $4$ .

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Step 2.3.2.4.2.1

Factor $2$ out of $- 46$ .

$y = - \frac{11 x}{16} + \frac{2 (- 23)}{4}$

Step 2.3.2.4.2.2

Cancel the common factors.

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Step 2.3.2.4.2.2.1

Factor $2$ out of $4$ .

$y = - \frac{11 x}{16} + \frac{2 \cdot - 23}{2 \cdot 2}$

Step 2.3.2.4.2.2.2

Cancel the common factor.

$y = - \frac{11 x}{16} + \frac{2 \cdot - 23}{2 \cdot 2}$

Step 2.3.2.4.2.2.3

Rewrite the expression.

$y = - \frac{11 x}{16} + \frac{- 23}{2}$

Step 2.3.2.4.3

Move the negative in front of the fraction.

$y = - \frac{11 x}{16} - \frac{23}{2}$

Step 2.3.3

Write in $y = m x + b$ form.

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Step 2.3.3.1

Reorder terms.

$y = - (\frac{11}{16} x) - \frac{23}{2}$

Step 2.3.3.2

Remove parentheses.

$y = - \frac{11}{16} x - \frac{23}{2}$

Step 3