Calculus Examples

$y = 3 x^{2} - 9 x + 7$ , $(2, 1)$

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 $x$ .

$\frac{d}{d x} (3 x^{2} - 9 x + 7)$

Step 2

By the Sum Rule, the derivative of $3 x^{2} - 9 x + 7$ with respect to $x$ is $\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- 9 x] + \frac{d}{d x} [7]$ .

$\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- 9 x] + \frac{d}{d x} [7]$

Step 3

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

Tap for more steps...

Step 3.1

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

$3 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 9 x] + \frac{d}{d x} [7]$

Step 3.2

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

$3 (2 x) + \frac{d}{d x} [- 9 x] + \frac{d}{d x} [7]$

Step 3.3

Multiply $2$ by $3$ .

$6 x + \frac{d}{d x} [- 9 x] + \frac{d}{d x} [7]$

Step 4

Evaluate $\frac{d}{d x} [- 9 x]$ .

Tap for more steps...

Step 4.1

Since $- 9$ is constant with respect to $x$ , the derivative of $- 9 x$ with respect to $x$ is $- 9 \frac{d}{d x} [x]$ .

$6 x - 9 \frac{d}{d x} [x] + \frac{d}{d x} [7]$

Step 4.2

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

$6 x - 9 \cdot 1 + \frac{d}{d x} [7]$

Step 4.3

Multiply $- 9$ by $1$ .

$6 x - 9 + \frac{d}{d x} [7]$

Step 5

Differentiate using the Constant Rule.

Tap for more steps...

Step 5.1

Since $7$ is constant with respect to $x$ , the derivative of $7$ with respect to $x$ is $0$ .

$6 x - 9 + 0$

Step 5.2

Add $6 x - 9$ and $0$ .

$6 x - 9$

Step 6

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

$f' (x) = 6 x - 9$

Step 7

Replace the variable $x$ with $2$ in the expression.

$f' (2) = 6 (2) - 9$

Step 8

Multiply $6$ by $2$ .

$f' (2) = 12 - 9$

Step 9

Subtract $9$ from $12$ .

$f' (2) = 3$

Step 10

The derivative at $(2, 1)$ is $3$ .

$3$