Calculus Examples

$y = {(3 x^{2} - 6 x + 1)}^{2}$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = 3 x^{2} - 6 x + 1$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $3 x^{2} - 6 x + 1$ .

$\frac{d}{d u} [u^{2}] \frac{d}{d x} [3 x^{2} - 6 x + 1]$

Step 1.2

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

$2 u \frac{d}{d x} [3 x^{2} - 6 x + 1]$

Step 1.3

Replace all occurrences of $u$ with $3 x^{2} - 6 x + 1$ .

$2 (3 x^{2} - 6 x + 1) \frac{d}{d x} [3 x^{2} - 6 x + 1]$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $3 x^{2} - 6 x + 1$ with respect to $x$ is $\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- 6 x] + \frac{d}{d x} [1]$ .

$2 (3 x^{2} - 6 x + 1) (\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- 6 x] + \frac{d}{d x} [1])$

Step 2.2

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

$2 (3 x^{2} - 6 x + 1) (3 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 6 x] + \frac{d}{d x} [1])$

Step 2.3

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

$2 (3 x^{2} - 6 x + 1) (3 (2 x) + \frac{d}{d x} [- 6 x] + \frac{d}{d x} [1])$

Step 2.4

Multiply $2$ by $3$ .

$2 (3 x^{2} - 6 x + 1) (6 x + \frac{d}{d x} [- 6 x] + \frac{d}{d x} [1])$

Step 2.5

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6 x$ with respect to $x$ is $- 6 \frac{d}{d x} [x]$ .

$2 (3 x^{2} - 6 x + 1) (6 x - 6 \frac{d}{d x} [x] + \frac{d}{d x} [1])$

Step 2.6

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

$2 (3 x^{2} - 6 x + 1) (6 x - 6 \cdot 1 + \frac{d}{d x} [1])$

Step 2.7

Multiply $- 6$ by $1$ .

$2 (3 x^{2} - 6 x + 1) (6 x - 6 + \frac{d}{d x} [1])$

Step 2.8

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$2 (3 x^{2} - 6 x + 1) (6 x - 6 + 0)$

Step 2.9

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

$2 (3 x^{2} - 6 x + 1) (6 x - 6)$

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$(2 (3 x^{2}) + 2 (- 6 x) + 2 \cdot 1) (6 x - 6)$

Step 3.2

Combine terms.

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Step 3.2.1

Multiply $3$ by $2$ .

$(6 x^{2} + 2 (- 6 x) + 2 \cdot 1) (6 x - 6)$

Step 3.2.2

Multiply $- 6$ by $2$ .

$(6 x^{2} - 12 x + 2 \cdot 1) (6 x - 6)$

Step 3.2.3

Multiply $2$ by $1$ .

$(6 x^{2} - 12 x + 2) (6 x - 6)$

Step 3.3

Reorder the factors of $(6 x^{2} - 12 x + 2) (6 x - 6)$ .

$(6 x - 6) (6 x^{2} - 12 x + 2)$