Calculus Examples

$f (x) = 2 x + \frac{5}{x}$ , $x = 4$

Step 1

Find the corresponding $y$ -value to $x = 4$ .

Tap for more steps...

Step 1.1

Substitute $4$ in for $x$ .

$y = 2 (4) + \frac{5}{4}$

Step 1.2

Solve for $y$ .

Tap for more steps...

Step 1.2.1

Remove parentheses.

$y = 2 (4) + \frac{5}{4}$

Step 1.2.2

Simplify $2 (4) + \frac{5}{4}$ .

Tap for more steps...

Step 1.2.2.1

Multiply $2$ by $4$ .

$y = 8 + \frac{5}{4}$

Step 1.2.2.2

To write $8$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$y = 8 \cdot \frac{4}{4} + \frac{5}{4}$

Step 1.2.2.3

Combine $8$ and $\frac{4}{4}$ .

$y = \frac{8 \cdot 4}{4} + \frac{5}{4}$

Step 1.2.2.4

Combine the numerators over the common denominator.

$y = \frac{8 \cdot 4 + 5}{4}$

Step 1.2.2.5

Simplify the numerator.

Tap for more steps...

Step 1.2.2.5.1

Multiply $8$ by $4$ .

$y = \frac{32 + 5}{4}$

Step 1.2.2.5.2

Add $32$ and $5$ .

$y = \frac{37}{4}$

Step 2

Find the first derivative and evaluate at $x = 4$ and $y = \frac{37}{4}$ to find the slope of the tangent line.

Tap for more steps...

Step 2.1

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

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

Step 2.2

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

Tap for more steps...

Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

Multiply $2$ by $1$ .

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

Step 2.3

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

Tap for more steps...

Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

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

$2 + 5 (- x^{- 2})$

Step 2.3.4

Multiply $- 1$ by $5$ .

$2 - 5 x^{- 2}$

Step 2.4

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

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

Step 2.5

Simplify.

Tap for more steps...

Step 2.5.1

Combine terms.

Tap for more steps...

Step 2.5.1.1

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

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

Step 2.5.1.2

Move the negative in front of the fraction.

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

Step 2.5.2

Reorder terms.

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

Step 2.6

Evaluate the derivative at $x = 4$ .

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

Step 2.7

Simplify.

Tap for more steps...

Step 2.7.1

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

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

Step 2.7.2

To write $2$ as a fraction with a common denominator, multiply by $\frac{16}{16}$ .

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

Step 2.7.3

Combine $2$ and $\frac{16}{16}$ .

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

Step 2.7.4

Combine the numerators over the common denominator.

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

Step 2.7.5

Simplify the numerator.

Tap for more steps...

Step 2.7.5.1

Multiply $2$ by $16$ .

$\frac{- 5 + 32}{16}$

Step 2.7.5.2

Add $- 5$ and $32$ .

$\frac{27}{16}$

Step 3

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

Tap for more steps...

Step 3.1

Use the slope $\frac{27}{16}$ and a given point $(4, \frac{37}{4})$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (\frac{37}{4}) = \frac{27}{16} \cdot (x - (4))$

Step 3.2

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

$y - \frac{37}{4} = \frac{27}{16} \cdot (x - 4)$

Step 3.3

Solve for $y$ .

Tap for more steps...

Step 3.3.1

Simplify $\frac{27}{16} \cdot (x - 4)$ .

Tap for more steps...

Step 3.3.1.1

Rewrite.

$y - \frac{37}{4} = 0 + 0 + \frac{27}{16} \cdot (x - 4)$

Step 3.3.1.2

Simplify by adding zeros.

$y - \frac{37}{4} = \frac{27}{16} \cdot (x - 4)$

Step 3.3.1.3

Apply the distributive property.

$y - \frac{37}{4} = \frac{27}{16} x + \frac{27}{16} \cdot - 4$

Step 3.3.1.4

Combine $\frac{27}{16}$ and $x$ .

$y - \frac{37}{4} = \frac{27 x}{16} + \frac{27}{16} \cdot - 4$

Step 3.3.1.5

Cancel the common factor of $4$ .

Tap for more steps...

Step 3.3.1.5.1

Factor $4$ out of $16$ .

$y - \frac{37}{4} = \frac{27 x}{16} + \frac{27}{4 (4)} \cdot - 4$

Step 3.3.1.5.2

Factor $4$ out of $- 4$ .

$y - \frac{37}{4} = \frac{27 x}{16} + \frac{27}{4 \cdot 4} \cdot (4 \cdot - 1)$

Step 3.3.1.5.3

Cancel the common factor.

$y - \frac{37}{4} = \frac{27 x}{16} + \frac{27}{4 \cdot 4} \cdot (4 \cdot - 1)$

Step 3.3.1.5.4

Rewrite the expression.

$y - \frac{37}{4} = \frac{27 x}{16} + \frac{27}{4} \cdot - 1$

Step 3.3.1.6

Combine $\frac{27}{4}$ and $- 1$ .

$y - \frac{37}{4} = \frac{27 x}{16} + \frac{27 \cdot - 1}{4}$

Step 3.3.1.7

Simplify the expression.

Tap for more steps...

Step 3.3.1.7.1

Multiply $27$ by $- 1$ .

$y - \frac{37}{4} = \frac{27 x}{16} + \frac{- 27}{4}$

Step 3.3.1.7.2

Move the negative in front of the fraction.

$y - \frac{37}{4} = \frac{27 x}{16} - \frac{27}{4}$

Step 3.3.2

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

Tap for more steps...

Step 3.3.2.1

Add $\frac{37}{4}$ to both sides of the equation.

$y = \frac{27 x}{16} - \frac{27}{4} + \frac{37}{4}$

Step 3.3.2.2

Combine the numerators over the common denominator.

$y = \frac{27 x}{16} + \frac{- 27 + 37}{4}$

Step 3.3.2.3

Add $- 27$ and $37$ .

$y = \frac{27 x}{16} + \frac{10}{4}$

Step 3.3.2.4

Cancel the common factor of $10$ and $4$ .

Tap for more steps...

Step 3.3.2.4.1

Factor $2$ out of $10$ .

$y = \frac{27 x}{16} + \frac{2 (5)}{4}$

Step 3.3.2.4.2

Cancel the common factors.

Tap for more steps...

Step 3.3.2.4.2.1

Factor $2$ out of $4$ .

$y = \frac{27 x}{16} + \frac{2 \cdot 5}{2 \cdot 2}$

Step 3.3.2.4.2.2

Cancel the common factor.

$y = \frac{27 x}{16} + \frac{2 \cdot 5}{2 \cdot 2}$

Step 3.3.2.4.2.3

Rewrite the expression.

$y = \frac{27 x}{16} + \frac{5}{2}$

Step 3.3.3

Reorder terms.

$y = \frac{27}{16} x + \frac{5}{2}$

Step 4