Calculus Examples

$y = x^{3} - 4 x$ , $(2, 0)$

Step 1

Find the derivative of the function. To find the slope of the equation tangent to the line, evaluate the derivative at the desired value of $x$ .

$\frac{d}{d x} (x^{3} - 4 x)$

Step 2

Differentiate.

Tap for more steps...

Step 2.1

By the Sum Rule, the derivative of $x^{3} - 4 x$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 4 x]$ .

$\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 4 x]$

Step 2.2

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

$3 x^{2} + \frac{d}{d x} [- 4 x]$

Step 3

Evaluate $\frac{d}{d x} [- 4 x]$ .

Tap for more steps...

Step 3.1

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4 x$ with respect to $x$ is $- 4 \frac{d}{d x} [x]$ .

$3 x^{2} - 4 \frac{d}{d x} [x]$

Step 3.2

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

$3 x^{2} - 4 \cdot 1$

Step 3.3

Multiply $- 4$ by $1$ .

$3 x^{2} - 4$

Step 4

The derivative of the equation in terms of $y$ can also be represented as $f' (x)$ .

$f' (x) = 3 x^{2} - 4$

Step 5

Replace the variable $x$ with $2$ in the expression.

$f' (2) = 3 {(2)}^{2} - 4$

Step 6

Simplify each term.

Tap for more steps...

Step 6.1

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

$f' (2) = 3 \cdot 4 - 4$

Step 6.2

Multiply $3$ by $4$ .

$f' (2) = 12 - 4$

Step 7

Subtract $4$ from $12$ .

$f' (2) = 8$

Step 8

The derivative at $(2, 0)$ is $8$ .

$8$