Calculus Examples

$y = 7 x^{2} + 5 x$ , $(- 1, 2)$

Step 1

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

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Step 1.1

By the Sum Rule, the derivative of $7 x^{2} + 5 x$ with respect to $x$ is $\frac{d}{d x} [7 x^{2}] + \frac{d}{d x} [5 x]$ .

$\frac{d}{d x} [7 x^{2}] + \frac{d}{d x} [5 x]$

Step 1.2

Evaluate $\frac{d}{d x} [7 x^{2}]$ .

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Step 1.2.1

Since $7$ is constant with respect to $x$ , the derivative of $7 x^{2}$ with respect to $x$ is $7 \frac{d}{d x} [x^{2}]$ .

$7 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [5 x]$

Step 1.2.2

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

$7 (2 x) + \frac{d}{d x} [5 x]$

Step 1.2.3

Multiply $2$ by $7$ .

$14 x + \frac{d}{d x} [5 x]$

Step 1.3

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

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Step 1.3.1

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

$14 x + 5 \frac{d}{d x} [x]$

Step 1.3.2

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

$14 x + 5 \cdot 1$

Step 1.3.3

Multiply $5$ by $1$ .

$14 x + 5$

Step 1.4

Evaluate the derivative at $x = - 1$ .

$14 (- 1) + 5$

Step 1.5

Simplify.

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Step 1.5.1

Multiply $14$ by $- 1$ .

$- 14 + 5$

Step 1.5.2

Add $- 14$ and $5$ .

$- 9$

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 $- 9$ and a given point $(- 1, 2)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (2) = - 9 \cdot (x - (- 1))$

Step 2.2

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

$y - 2 = - 9 \cdot (x + 1)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $- 9 \cdot (x + 1)$ .

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Step 2.3.1.1

Rewrite.

$y - 2 = 0 + 0 - 9 \cdot (x + 1)$

Step 2.3.1.2

Simplify by adding zeros.

$y - 2 = - 9 \cdot (x + 1)$

Step 2.3.1.3

Apply the distributive property.

$y - 2 = - 9 x - 9 \cdot 1$

Step 2.3.1.4

Multiply $- 9$ by $1$ .

$y - 2 = - 9 x - 9$

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 $2$ to both sides of the equation.

$y = - 9 x - 9 + 2$

Step 2.3.2.2

Add $- 9$ and $2$ .

$y = - 9 x - 7$

Step 3