Calculus Examples

y = x^{3} - 2 x

$y = x^{3} - 2 x$ ,

(2, 4)

$(2, 4)$

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

$x$ .

\frac{d}{d x} (x^{3} - 2 x)

$\frac{d}{d x} (x^{3} - 2 x)$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of

x^{3} - 2 x

$x^{3} - 2 x$ with respect to

x

$x$ is

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

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

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

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

Step 2.2

Differentiate using the Power Rule which states that

\frac{d}{d x} [x^{n}]

$\frac{d}{d x} [x^{n}]$ is

n x^{n - 1}

$n x^{n - 1}$ where

n = 3

$n = 3$ .

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

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

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

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

Step 3

Evaluate

\frac{d}{d x} [- 2 x]

$\frac{d}{d x} [- 2 x]$ .

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Step 3.1

Since

- 2

$- 2$ is constant with respect to

x

$x$ , the derivative of

- 2 x

$- 2 x$ with respect to

x

$x$ is

- 2 \frac{d}{d x} [x]

$- 2 \frac{d}{d x} [x]$ .

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

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

Step 3.2

Differentiate using the Power Rule which states that

\frac{d}{d x} [x^{n}]

$\frac{d}{d x} [x^{n}]$ is

n x^{n - 1}

$n x^{n - 1}$ where

n = 1

$n = 1$ .

3 x^{2} - 2 \cdot 1

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

Step 3.3

Multiply

- 2

$- 2$ by

1

$1$ .

3 x^{2} - 2

$3 x^{2} - 2$

3 x^{2} - 2

$3 x^{2} - 2$

Step 4

The derivative of the equation in terms of

y

$y$ can also be represented as

$f' (x)$ .

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

Step 5

Replace the variable

$x$ with

$2$ in the expression.

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

Step 6

Simplify each term.

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Step 6.1

Raise

$2$ to the power of

$2$ .

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

Step 6.2

Multiply

$3$ by

$4$ .

$f' (2) = 12 - 2$

Step 7

Subtract

$2$ from

$12$ .

$f' (2) = 10$

Step 8

The derivative at

$(2, 4)$ is

$10$ .

$10$