Calculus Examples

$f (x) = \sqrt{4 x^{2} - 12 x + 9}$

Step 1

Rewrite $\frac{d}{d x} [\sqrt{4 x^{2} - 12 x + 9}]$ as $\frac{d}{d x} [2 x - 3]$ .

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Step 1.1

Factor using the perfect square rule.

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Step 1.1.1

Rewrite $4 x^{2}$ as ${(2 x)}^{2}$ .

$\frac{d}{d x} [\sqrt{{(2 x)}^{2} - 12 x + 9}]$

Step 1.1.2

Rewrite $9$ as $3^{2}$ .

$\frac{d}{d x} [\sqrt{{(2 x)}^{2} - 12 x + 3^{2}}]$

Step 1.1.3

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$12 x = 2 \cdot (2 x) \cdot 3$

Step 1.1.4

Rewrite the polynomial.

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

Step 1.1.5

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = 2 x$ and $b = 3$ .

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

Step 1.2

Pull terms out from under the radical, assuming positive real numbers.

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

Step 2

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

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

Step 3

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

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

Step 4

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

$2 \cdot 1 + \frac{d}{d x} [- 3]$

Step 5

Multiply $2$ by $1$ .

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

Step 6

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

$2 + 0$

Step 7

Add $2$ and $0$ .

$2$