Calculus Examples

$y = \frac{6 x}{1 + 11 x^{2}}$ , $(1, \frac{1}{2})$

Step 1

Find the first derivative and evaluate at $x = 1$ and $y = \frac{1}{2}$ to find the slope of the tangent line.

Tap for more steps...

Step 1.1

Since $6$ is constant with respect to $x$ , the derivative of $\frac{6 x}{1 + 11 x^{2}}$ with respect to $x$ is $6 \frac{d}{d x} [\frac{x}{1 + 11 x^{2}}]$ .

$6 \frac{d}{d x} [\frac{x}{1 + 11 x^{2}}]$

Step 1.2

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = x$ and $g (x) = 1 + 11 x^{2}$ .

$6 \frac{(1 + 11 x^{2}) \frac{d}{d x} [x] - x \frac{d}{d x} [1 + 11 x^{2}]}{{(1 + 11 x^{2})}^{2}}$

Step 1.3

Differentiate.

Tap for more steps...

Step 1.3.1

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

$6 \frac{(1 + 11 x^{2}) \cdot 1 - x \frac{d}{d x} [1 + 11 x^{2}]}{{(1 + 11 x^{2})}^{2}}$

Step 1.3.2

Multiply $1 + 11 x^{2}$ by $1$ .

$6 \frac{1 + 11 x^{2} - x \frac{d}{d x} [1 + 11 x^{2}]}{{(1 + 11 x^{2})}^{2}}$

Step 1.3.3

By the Sum Rule, the derivative of $1 + 11 x^{2}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [11 x^{2}]$ .

$6 \frac{1 + 11 x^{2} - x (\frac{d}{d x} [1] + \frac{d}{d x} [11 x^{2}])}{{(1 + 11 x^{2})}^{2}}$

Step 1.3.4

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

$6 \frac{1 + 11 x^{2} - x (0 + \frac{d}{d x} [11 x^{2}])}{{(1 + 11 x^{2})}^{2}}$

Step 1.3.5

Add $0$ and $\frac{d}{d x} [11 x^{2}]$ .

$6 \frac{1 + 11 x^{2} - x \frac{d}{d x} [11 x^{2}]}{{(1 + 11 x^{2})}^{2}}$

Step 1.3.6

Since $11$ is constant with respect to $x$ , the derivative of $11 x^{2}$ with respect to $x$ is $11 \frac{d}{d x} [x^{2}]$ .

$6 \frac{1 + 11 x^{2} - x (11 \frac{d}{d x} [x^{2}])}{{(1 + 11 x^{2})}^{2}}$

Step 1.3.7

Multiply $11$ by $- 1$ .

$6 \frac{1 + 11 x^{2} - 11 x \frac{d}{d x} [x^{2}]}{{(1 + 11 x^{2})}^{2}}$

Step 1.3.8

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

$6 \frac{1 + 11 x^{2} - 11 x (2 x)}{{(1 + 11 x^{2})}^{2}}$

Step 1.3.9

Multiply $2$ by $- 11$ .

$6 \frac{1 + 11 x^{2} - 22 x \cdot x}{{(1 + 11 x^{2})}^{2}}$

Step 1.4

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

$6 \frac{1 + 11 x^{2} - 22 (x^{1} x)}{{(1 + 11 x^{2})}^{2}}$

Step 1.5

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

$6 \frac{1 + 11 x^{2} - 22 (x^{1} x^{1})}{{(1 + 11 x^{2})}^{2}}$

Step 1.6

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

$6 \frac{1 + 11 x^{2} - 22 x^{1 + 1}}{{(1 + 11 x^{2})}^{2}}$

Step 1.7

Add $1$ and $1$ .

$6 \frac{1 + 11 x^{2} - 22 x^{2}}{{(1 + 11 x^{2})}^{2}}$

Step 1.8

Subtract $22 x^{2}$ from $11 x^{2}$ .

$6 \frac{1 - 11 x^{2}}{{(1 + 11 x^{2})}^{2}}$

Step 1.9

Combine $6$ and $\frac{1 - 11 x^{2}}{{(1 + 11 x^{2})}^{2}}$ .

$\frac{6 (1 - 11 x^{2})}{{(1 + 11 x^{2})}^{2}}$

Step 1.10

Simplify.

Tap for more steps...

Step 1.10.1

Apply the distributive property.

$\frac{6 \cdot 1 + 6 (- 11 x^{2})}{{(1 + 11 x^{2})}^{2}}$

Step 1.10.2

Simplify each term.

Tap for more steps...

Step 1.10.2.1

Multiply $6$ by $1$ .

$\frac{6 + 6 (- 11 x^{2})}{{(1 + 11 x^{2})}^{2}}$

Step 1.10.2.2

Multiply $- 11$ by $6$ .

$\frac{6 - 66 x^{2}}{{(1 + 11 x^{2})}^{2}}$

Step 1.10.3

Reorder terms.

$\frac{- 66 x^{2} + 6}{{(11 x^{2} + 1)}^{2}}$

Step 1.11

Evaluate the derivative at $x = 1$ .

$\frac{- 66 {(1)}^{2} + 6}{{(11 {(1)}^{2} + 1)}^{2}}$

Step 1.12

Simplify.

Tap for more steps...

Step 1.12.1

Simplify the numerator.

Tap for more steps...

Step 1.12.1.1

One to any power is one.

$\frac{- 66 \cdot 1 + 6}{{(11 \cdot 1^{2} + 1)}^{2}}$

Step 1.12.1.2

Multiply $- 66$ by $1$ .

$\frac{- 66 + 6}{{(11 \cdot 1^{2} + 1)}^{2}}$

Step 1.12.1.3

Add $- 66$ and $6$ .

$\frac{- 60}{{(11 \cdot 1^{2} + 1)}^{2}}$

Step 1.12.2

Simplify the denominator.

Tap for more steps...

Step 1.12.2.1

One to any power is one.

$\frac{- 60}{{(11 \cdot 1 + 1)}^{2}}$

Step 1.12.2.2

Multiply $11$ by $1$ .

$\frac{- 60}{{(11 + 1)}^{2}}$

Step 1.12.2.3

Add $11$ and $1$ .

$\frac{- 60}{12^{2}}$

Step 1.12.2.4

Raise $12$ to the power of $2$ .

$\frac{- 60}{144}$

Step 1.12.3

Reduce the expression by cancelling the common factors.

Tap for more steps...

Step 1.12.3.1

Cancel the common factor of $- 60$ and $144$ .

Tap for more steps...

Step 1.12.3.1.1

Factor $12$ out of $- 60$ .

$\frac{12 (- 5)}{144}$

Step 1.12.3.1.2

Cancel the common factors.

Tap for more steps...

Step 1.12.3.1.2.1

Factor $12$ out of $144$ .

$\frac{12 \cdot - 5}{12 \cdot 12}$

Step 1.12.3.1.2.2

Cancel the common factor.

$\frac{12 \cdot - 5}{12 \cdot 12}$

Step 1.12.3.1.2.3

Rewrite the expression.

$\frac{- 5}{12}$

Step 1.12.3.2

Move the negative in front of the fraction.

$- \frac{5}{12}$

Step 2

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

Tap for more steps...

Step 2.1

Use the slope $- \frac{5}{12}$ and a given point $(1, \frac{1}{2})$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (\frac{1}{2}) = - \frac{5}{12} \cdot (x - (1))$

Step 2.2

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

$y - \frac{1}{2} = - \frac{5}{12} \cdot (x - 1)$

Step 2.3

Solve for $y$ .

Tap for more steps...

Step 2.3.1

Simplify $- \frac{5}{12} \cdot (x - 1)$ .

Tap for more steps...

Step 2.3.1.1

Rewrite.

$y - \frac{1}{2} = 0 + 0 - \frac{5}{12} \cdot (x - 1)$

Step 2.3.1.2

Simplify by adding zeros.

$y - \frac{1}{2} = - \frac{5}{12} \cdot (x - 1)$

Step 2.3.1.3

Apply the distributive property.

$y - \frac{1}{2} = - \frac{5}{12} x - \frac{5}{12} \cdot - 1$

Step 2.3.1.4

Combine $x$ and $\frac{5}{12}$ .

$y - \frac{1}{2} = - \frac{x \cdot 5}{12} - \frac{5}{12} \cdot - 1$

Step 2.3.1.5

Multiply $- \frac{5}{12} \cdot - 1$ .

Tap for more steps...

Step 2.3.1.5.1

Multiply $- 1$ by $- 1$ .

$y - \frac{1}{2} = - \frac{x \cdot 5}{12} + 1 (\frac{5}{12})$

Step 2.3.1.5.2

Multiply $\frac{5}{12}$ by $1$ .

$y - \frac{1}{2} = - \frac{x \cdot 5}{12} + \frac{5}{12}$

Step 2.3.1.6

Move $5$ to the left of $x$ .

$y - \frac{1}{2} = - \frac{5 x}{12} + \frac{5}{12}$

Step 2.3.2

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

Tap for more steps...

Step 2.3.2.1

Add $\frac{1}{2}$ to both sides of the equation.

$y = - \frac{5 x}{12} + \frac{5}{12} + \frac{1}{2}$

Step 2.3.2.2

To write $\frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$y = - \frac{5 x}{12} + \frac{5}{12} + \frac{1}{2} \cdot \frac{6}{6}$

Step 2.3.2.3

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 2.3.2.3.1

Multiply $\frac{1}{2}$ by $\frac{6}{6}$ .

$y = - \frac{5 x}{12} + \frac{5}{12} + \frac{6}{2 \cdot 6}$

Step 2.3.2.3.2

Multiply $2$ by $6$ .

$y = - \frac{5 x}{12} + \frac{5}{12} + \frac{6}{12}$

Step 2.3.2.4

Combine the numerators over the common denominator.

$y = - \frac{5 x}{12} + \frac{5 + 6}{12}$

Step 2.3.2.5

Add $5$ and $6$ .

$y = - \frac{5 x}{12} + \frac{11}{12}$

Step 2.3.3

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

Tap for more steps...

Step 2.3.3.1

Reorder terms.

$y = - (\frac{5}{12} x) + \frac{11}{12}$

Step 2.3.3.2

Remove parentheses.

$y = - \frac{5}{12} x + \frac{11}{12}$

Step 3