Calculus Examples

$y = 5 x^{3} - 5 x$ at the point $(- 1, 0)$

Step 1

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

Tap for more steps...

Step 1.1

By the Sum Rule, the derivative of $5 x^{3} - 5 x$ with respect to $x$ is $\frac{d}{d x} [5 x^{3}] + \frac{d}{d x} [- 5 x]$ .

$\frac{d}{d x} [5 x^{3}] + \frac{d}{d x} [- 5 x]$

Step 1.2

Evaluate $\frac{d}{d x} [5 x^{3}]$ .

Tap for more steps...

Step 1.2.1

Since $5$ is constant with respect to $x$ , the derivative of $5 x^{3}$ with respect to $x$ is $5 \frac{d}{d x} [x^{3}]$ .

$5 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 5 x]$

Step 1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$5 (3 x^{2}) + \frac{d}{d x} [- 5 x]$

Step 1.2.3

Multiply $3$ by $5$ .

$15 x^{2} + \frac{d}{d x} [- 5 x]$

Step 1.3

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

Tap for more steps...

Step 1.3.1

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

$15 x^{2} - 5 \frac{d}{d x} [x]$

Step 1.3.2

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

$15 x^{2} - 5 \cdot 1$

Step 1.3.3

Multiply $- 5$ by $1$ .

$15 x^{2} - 5$

Step 1.4

Evaluate the derivative at $x = - 1$ .

$15 {(- 1)}^{2} - 5$

Step 1.5

Simplify.

Tap for more steps...

Step 1.5.1

Simplify each term.

Tap for more steps...

Step 1.5.1.1

Raise $- 1$ to the power of $2$ .

$15 \cdot 1 - 5$

Step 1.5.1.2

Multiply $15$ by $1$ .

$15 - 5$

Step 1.5.2

Subtract $5$ from $15$ .

$10$

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 $10$ and a given point $(- 1, 0)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (0) = 10 \cdot (x - (- 1))$

Step 2.2

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

$y + 0 = 10 \cdot (x + 1)$

Step 2.3

Solve for $y$ .

Tap for more steps...

Step 2.3.1

Add $y$ and $0$ .

$y = 10 \cdot (x + 1)$

Step 2.3.2

Simplify $10 \cdot (x + 1)$ .

Tap for more steps...

Step 2.3.2.1

Apply the distributive property.

$y = 10 x + 10 \cdot 1$

Step 2.3.2.2

Multiply $10$ by $1$ .

$y = 10 x + 10$

Step 3