Calculus Examples

$f (x) = {(6 x + 1)}^{2}$ ;, $(0, 1)$

Step 1

Find the first derivative and evaluate at $x = 0$ and $y = 1$ to find the slope of the tangent line.

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Step 1.1

Rewrite ${(6 x + 1)}^{2}$ as $(6 x + 1) (6 x + 1)$ .

$\frac{d}{d x} [(6 x + 1) (6 x + 1)]$

Step 1.2

Expand $(6 x + 1) (6 x + 1)$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

$\frac{d}{d x} [6 x (6 x + 1) + 1 (6 x + 1)]$

Step 1.2.2

Apply the distributive property.

$\frac{d}{d x} [6 x (6 x) + 6 x \cdot 1 + 1 (6 x + 1)]$

Step 1.2.3

Apply the distributive property.

$\frac{d}{d x} [6 x (6 x) + 6 x \cdot 1 + 1 (6 x) + 1 \cdot 1]$

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Rewrite using the commutative property of multiplication.

$\frac{d}{d x} [6 \cdot 6 x \cdot x + 6 x \cdot 1 + 1 (6 x) + 1 \cdot 1]$

Step 1.3.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 1.3.1.2.1

Move $x$ .

$\frac{d}{d x} [6 \cdot 6 (x \cdot x) + 6 x \cdot 1 + 1 (6 x) + 1 \cdot 1]$

Step 1.3.1.2.2

Multiply $x$ by $x$ .

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

Step 1.3.1.3

Multiply $6$ by $6$ .

$\frac{d}{d x} [36 x^{2} + 6 x \cdot 1 + 1 (6 x) + 1 \cdot 1]$

Step 1.3.1.4

Multiply $6$ by $1$ .

$\frac{d}{d x} [36 x^{2} + 6 x + 1 (6 x) + 1 \cdot 1]$

Step 1.3.1.5

Multiply $6 x$ by $1$ .

$\frac{d}{d x} [36 x^{2} + 6 x + 6 x + 1 \cdot 1]$

Step 1.3.1.6

Multiply $1$ by $1$ .

$\frac{d}{d x} [36 x^{2} + 6 x + 6 x + 1]$

Step 1.3.2

Add $6 x$ and $6 x$ .

$\frac{d}{d x} [36 x^{2} + 12 x + 1]$

Step 1.4

By the Sum Rule, the derivative of $36 x^{2} + 12 x + 1$ with respect to $x$ is $\frac{d}{d x} [36 x^{2}] + \frac{d}{d x} [12 x] + \frac{d}{d x} [1]$ .

$\frac{d}{d x} [36 x^{2}] + \frac{d}{d x} [12 x] + \frac{d}{d x} [1]$

Step 1.5

Since $36$ is constant with respect to $x$ , the derivative of $36 x^{2}$ with respect to $x$ is $36 \frac{d}{d x} [x^{2}]$ .

$36 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [12 x] + \frac{d}{d x} [1]$

Step 1.6

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

$36 (2 x) + \frac{d}{d x} [12 x] + \frac{d}{d x} [1]$

Step 1.7

Multiply $2$ by $36$ .

$72 x + \frac{d}{d x} [12 x] + \frac{d}{d x} [1]$

Step 1.8

Since $12$ is constant with respect to $x$ , the derivative of $12 x$ with respect to $x$ is $12 \frac{d}{d x} [x]$ .

$72 x + 12 \frac{d}{d x} [x] + \frac{d}{d x} [1]$

Step 1.9

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

$72 x + 12 \cdot 1 + \frac{d}{d x} [1]$

Step 1.10

Multiply $12$ by $1$ .

$72 x + 12 + \frac{d}{d x} [1]$

Step 1.11

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

$72 x + 12 + 0$

Step 1.12

Add $72 x + 12$ and $0$ .

$72 x + 12$

Step 1.13

Evaluate the derivative at $x = 0$ .

$72 (0) + 12$

Step 1.14

Simplify.

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Step 1.14.1

Multiply $72$ by $0$ .

$0 + 12$

Step 1.14.2

Add $0$ and $12$ .

$12$

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 $12$ and a given point $(0, 1)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (1) = 12 \cdot (x - (0))$

Step 2.2

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

$y - 1 = 12 \cdot (x + 0)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Add $x$ and $0$ .

$y - 1 = 12 x$

Step 2.3.2

Add $1$ to both sides of the equation.

$y = 12 x + 1$

Step 3