Calculus Examples

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

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 $x$ .

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

Step 2

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

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

Step 3

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

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Step 3.1

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

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

Step 3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

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

Step 3.3

Multiply $2$ by $3$ .

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

Step 4

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

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Step 4.1

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

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

Step 4.2

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

$6 x - 2 \cdot 1$

Step 4.3

Multiply $- 2$ by $1$ .

$6 x - 2$

Step 5

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

$f' (x) = 6 x - 2$

Step 6

Replace the variable $x$ with $- 1$ in the expression.

$f' (- 1) = 6 (- 1) - 2$

Step 7

Multiply $6$ by $- 1$ .

$f' (- 1) = - 6 - 2$

Step 8

Subtract $2$ from $- 6$ .

$f' (- 1) = - 8$

Step 9

The derivative at $(- 1, 5)$ is $- 8$ .

$- 8$