Calculus Examples

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

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 $x$ .

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

Step 2

Differentiate.

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Step 2.1

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

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

Step 2.2

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

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Multiply $- 2$ by $1$ .

$3 x^{2} - 2$

Step 4

The derivative of the equation in terms of $y$ 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 $10$ .

$10$