Calculus Examples

$y = 2 x^{3} - 5 x$ , $(1, - 3)$

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 $x$ .

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

Step 2

By the Sum Rule, the derivative of $2 x^{3} - 5 x$ with respect to $x$ is $\frac{d}{d x} [2 x^{3}] + \frac{d}{d x} [- 5 x]$ .

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

Step 3

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

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Step 3.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x^{3}$ with respect to $x$ is $2 \frac{d}{d x} [x^{3}]$ .

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

Step 3.2

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

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

Step 3.3

Multiply $3$ by $2$ .

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

Step 4

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

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

Multiply $- 5$ by $1$ .

$6 x^{2} - 5$

Step 5

The derivative of the equation in terms of $y$ 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 $1$ .

$1$