Calculus Examples

$y = x^{2} + 6$ , $(1, 7)$

Step 1

Find the first derivative and evaluate at $x = 1$ and $y = 7$ to find the slope of the tangent line.

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Step 1.1

By the Sum Rule, the derivative of $x^{2} + 6$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [6]$ .

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

Step 1.2

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

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

Step 1.3

Since $6$ is constant with respect to $x$ , the derivative of $6$ with respect to $x$ is $0$ .

$2 x + 0$

Step 1.4

Add $2 x$ and $0$ .

$2 x$

Step 1.5

Evaluate the derivative at $x = 1$ .

$2 (1)$

Step 1.6

Multiply $2$ by $1$ .

$2$

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 $2$ and a given point $(1, 7)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (7) = 2 \cdot (x - (1))$

Step 2.2

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

$y - 7 = 2 \cdot (x - 1)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $2 \cdot (x - 1)$ .

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Step 2.3.1.1

Rewrite.

$y - 7 = 0 + 0 + 2 \cdot (x - 1)$

Step 2.3.1.2

Simplify by adding zeros.

$y - 7 = 2 \cdot (x - 1)$

Step 2.3.1.3

Apply the distributive property.

$y - 7 = 2 x + 2 \cdot - 1$

Step 2.3.1.4

Multiply $2$ by $- 1$ .

$y - 7 = 2 x - 2$

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 $7$ to both sides of the equation.

$y = 2 x - 2 + 7$

Step 2.3.2.2

Add $- 2$ and $7$ .

$y = 2 x + 5$

Step 3