Calculus Examples

$f (x) = 4 x + x^{- \frac{1}{2}} + 1$ at the point $(4, \frac{35}{2})$

Step 1

Find the first derivative and evaluate at $x = 4$ and $y = \frac{35}{2}$ to find the slope of the tangent line.

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Step 1.1

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

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

Step 1.2

Evaluate $\frac{d}{d x} [4 x]$ .

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.2.3

Multiply $4$ by $1$ .

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

Step 1.3

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

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Step 1.3.1

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

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

Step 1.3.2

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

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

Step 1.3.3

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

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

Step 1.3.4

Combine the numerators over the common denominator.

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

Step 1.3.5

Simplify the numerator.

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Step 1.3.5.1

Multiply $- 1$ by $2$ .

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

Step 1.3.5.2

Subtract $2$ from $- 1$ .

$4 - \frac{1}{2} x^{\frac{- 3}{2}} + \frac{d}{d x} [1]$

Step 1.3.6

Move the negative in front of the fraction.

$4 - \frac{1}{2} x^{- \frac{3}{2}} + \frac{d}{d x} [1]$

Step 1.4

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$4 - \frac{1}{2} x^{- \frac{3}{2}} + 0$

Step 1.5

Simplify.

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Step 1.5.1

Add $4 - \frac{1}{2} x^{- \frac{3}{2}}$ and $0$ .

$4 - \frac{1}{2} x^{- \frac{3}{2}}$

Step 1.5.2

Reorder terms.

$- \frac{1}{2} x^{- \frac{3}{2}} + 4$

Step 1.5.3

Simplify each term.

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Step 1.5.3.1

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

$- \frac{1}{2} \cdot \frac{1}{x^{\frac{3}{2}}} + 4$

Step 1.5.3.2

Multiply $\frac{1}{x^{\frac{3}{2}}}$ by $\frac{1}{2}$ .

$- \frac{1}{x^{\frac{3}{2}} \cdot 2} + 4$

Step 1.5.3.3

Move $2$ to the left of $x^{\frac{3}{2}}$ .

$- \frac{1}{2 x^{\frac{3}{2}}} + 4$

Step 1.6

Evaluate the derivative at $x = 4$ .

$- \frac{1}{2 {(4)}^{\frac{3}{2}}} + 4$

Step 1.7

Simplify.

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Step 1.7.1

Simplify each term.

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Step 1.7.1.1

Simplify the denominator.

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Step 1.7.1.1.1

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

$- \frac{1}{2 \cdot {(2^{2})}^{\frac{3}{2}}} + 4$

Step 1.7.1.1.2

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

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Step 1.7.1.1.2.1

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

$- \frac{1}{2 \cdot 2^{2 (\frac{3}{2})}} + 4$

Step 1.7.1.1.2.2

Cancel the common factor of $2$ .

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Step 1.7.1.1.2.2.1

Cancel the common factor.

$- \frac{1}{2 \cdot 2^{2 (\frac{3}{2})}} + 4$

Step 1.7.1.1.2.2.2

Rewrite the expression.

$- \frac{1}{2 \cdot 2^{3}} + 4$

Step 1.7.1.1.3

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

$- \frac{1}{2^{1 + 3}} + 4$

Step 1.7.1.1.4

Add $1$ and $3$ .

$- \frac{1}{2^{4}} + 4$

Step 1.7.1.2

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

$- \frac{1}{16} + 4$

Step 1.7.2

To write $4$ as a fraction with a common denominator, multiply by $\frac{16}{16}$ .

$- \frac{1}{16} + 4 \cdot \frac{16}{16}$

Step 1.7.3

Combine $4$ and $\frac{16}{16}$ .

$- \frac{1}{16} + \frac{4 \cdot 16}{16}$

Step 1.7.4

Combine the numerators over the common denominator.

$\frac{- 1 + 4 \cdot 16}{16}$

Step 1.7.5

Simplify the numerator.

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Step 1.7.5.1

Multiply $4$ by $16$ .

$\frac{- 1 + 64}{16}$

Step 1.7.5.2

Add $- 1$ and $64$ .

$\frac{63}{16}$

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 $\frac{63}{16}$ and a given point $(4, \frac{35}{2})$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (\frac{35}{2}) = \frac{63}{16} \cdot (x - (4))$

Step 2.2

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

$y - \frac{35}{2} = \frac{63}{16} \cdot (x - 4)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $\frac{63}{16} \cdot (x - 4)$ .

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Step 2.3.1.1

Rewrite.

$y - \frac{35}{2} = 0 + 0 + \frac{63}{16} \cdot (x - 4)$

Step 2.3.1.2

Simplify by adding zeros.

$y - \frac{35}{2} = \frac{63}{16} \cdot (x - 4)$

Step 2.3.1.3

Apply the distributive property.

$y - \frac{35}{2} = \frac{63}{16} x + \frac{63}{16} \cdot - 4$

Step 2.3.1.4

Combine $\frac{63}{16}$ and $x$ .

$y - \frac{35}{2} = \frac{63 x}{16} + \frac{63}{16} \cdot - 4$

Step 2.3.1.5

Cancel the common factor of $4$ .

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Step 2.3.1.5.1

Factor $4$ out of $16$ .

$y - \frac{35}{2} = \frac{63 x}{16} + \frac{63}{4 (4)} \cdot - 4$

Step 2.3.1.5.2

Factor $4$ out of $- 4$ .

$y - \frac{35}{2} = \frac{63 x}{16} + \frac{63}{4 \cdot 4} \cdot (4 \cdot - 1)$

Step 2.3.1.5.3

Cancel the common factor.

$y - \frac{35}{2} = \frac{63 x}{16} + \frac{63}{4 \cdot 4} \cdot (4 \cdot - 1)$

Step 2.3.1.5.4

Rewrite the expression.

$y - \frac{35}{2} = \frac{63 x}{16} + \frac{63}{4} \cdot - 1$

Step 2.3.1.6

Combine $\frac{63}{4}$ and $- 1$ .

$y - \frac{35}{2} = \frac{63 x}{16} + \frac{63 \cdot - 1}{4}$

Step 2.3.1.7

Simplify the expression.

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Step 2.3.1.7.1

Multiply $63$ by $- 1$ .

$y - \frac{35}{2} = \frac{63 x}{16} + \frac{- 63}{4}$

Step 2.3.1.7.2

Move the negative in front of the fraction.

$y - \frac{35}{2} = \frac{63 x}{16} - \frac{63}{4}$

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

Add $\frac{35}{2}$ to both sides of the equation.

$y = \frac{63 x}{16} - \frac{63}{4} + \frac{35}{2}$

Step 2.3.2.2

To write $\frac{35}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = \frac{63 x}{16} - \frac{63}{4} + \frac{35}{2} \cdot \frac{2}{2}$

Step 2.3.2.3

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.3.2.3.1

Multiply $\frac{35}{2}$ by $\frac{2}{2}$ .

$y = \frac{63 x}{16} - \frac{63}{4} + \frac{35 \cdot 2}{2 \cdot 2}$

Step 2.3.2.3.2

Multiply $2$ by $2$ .

$y = \frac{63 x}{16} - \frac{63}{4} + \frac{35 \cdot 2}{4}$

Step 2.3.2.4

Combine the numerators over the common denominator.

$y = \frac{63 x}{16} + \frac{- 63 + 35 \cdot 2}{4}$

Step 2.3.2.5

Simplify the numerator.

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Step 2.3.2.5.1

Multiply $35$ by $2$ .

$y = \frac{63 x}{16} + \frac{- 63 + 70}{4}$

Step 2.3.2.5.2

Add $- 63$ and $70$ .

$y = \frac{63 x}{16} + \frac{7}{4}$

Step 2.3.3

Reorder terms.

$y = \frac{63}{16} x + \frac{7}{4}$

Step 3