Calculus Examples

y = 2 x^{3} - 5 x

$y = 2 x^{3} - 5 x$ ,

(1, - 3)

$(1, - 3)$

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} (2 x^{3} - 5 x)

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

Step 2

By the Sum Rule, the derivative of

2 x^{3} - 5 x

$2 x^{3} - 5 x$ with respect to

x

$x$ is

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

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

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

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

Step 3

Evaluate

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

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

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Step 3.1

Since

2

$2$ is constant with respect to

x

$x$ , the derivative of

2 x^{3}

$2 x^{3}$ with respect to

x

$x$ is

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

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

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

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

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 = 3

$n = 3$ .

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

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

Step 3.3

Multiply

3

$3$ by

2

$2$ .

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

$6 x^{2} + \frac{d}{d x} [- 5 x]$

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

$6 x^{2} + \frac{d}{d x} [- 5 x]$

Step 4

Evaluate

\frac{d}{d x} [- 5 x]

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

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Step 4.1

Since

- 5

$- 5$ is constant with respect to

x

$x$ , the derivative of

- 5 x

$- 5 x$ with respect to

x

$x$ is

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

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

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

$6 x^{2} - 5 \frac{d}{d x} [x]$

Step 4.2

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

$n = 1$ .

6 x^{2} - 5 \cdot 1

$6 x^{2} - 5 \cdot 1$

Step 4.3

Multiply

- 5

$- 5$ by

1

$1$ .

6 x^{2} - 5

$6 x^{2} - 5$

6 x^{2} - 5

$6 x^{2} - 5$

Step 5

The derivative of the equation in terms of

y

$y$ can also be represented as

$f' (x)$ .

$f' (x) = 6 x^{2} - 5$

Step 6

Replace the variable

$x$ with

$1$ in the expression.

$f' (1) = 6 {(1)}^{2} - 5$

Step 7

Simplify each term.

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Step 7.1

One to any power is one.

$f' (1) = 6 \cdot 1 - 5$

Step 7.2

Multiply

$6$ by

$1$ .

$f' (1) = 6 - 5$

Step 8

Subtract

$5$ from

$6$ .

$f' (1) = 1$

Step 9

The derivative at

$(1, - 3)$ is

$1$ .

$1$