Calculus Examples

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

Step 1

Find the first derivative.

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Step 1.1

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 1.1.1

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 1.1.2

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 1.1.3

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 1.2

Differentiate.

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Step 1.2.1

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 1.2.2

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 1.2.3

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 1.2.4

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 1.2.5

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 1.2.6

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 1.2.7

Multiply $- 6$ by $1$ .

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

Step 1.2.8

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 1.2.9

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

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

Step 1.3

Simplify.

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Step 1.3.1

Apply the distributive property.

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

Step 1.3.2

Combine terms.

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Step 1.3.2.1

Multiply $3$ by $2$ .

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

Step 1.3.2.2

Multiply $- 6$ by $2$ .

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

Step 1.3.2.3

Multiply $2$ by $1$ .

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

Step 1.3.3

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

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

Step 2

Find the second derivative.

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Step 2.1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = 6 x - 6$ and $g (x) = 6 x^{2} - 12 x + 2$ .

$(6 x - 6) \frac{d}{d x} [6 x^{2} - 12 x + 2] + (6 x^{2} - 12 x + 2) \frac{d}{d x} [6 x - 6]$

Step 2.2

Differentiate.

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Step 2.2.1

By the Sum Rule, the derivative of $6 x^{2} - 12 x + 2$ with respect to $x$ is $\frac{d}{d x} [6 x^{2}] + \frac{d}{d x} [- 12 x] + \frac{d}{d x} [2]$ .

$(6 x - 6) (\frac{d}{d x} [6 x^{2}] + \frac{d}{d x} [- 12 x] + \frac{d}{d x} [2]) + (6 x^{2} - 12 x + 2) \frac{d}{d x} [6 x - 6]$

Step 2.2.2

Since $6$ is constant with respect to $x$ , the derivative of $6 x^{2}$ with respect to $x$ is $6 \frac{d}{d x} [x^{2}]$ .

$(6 x - 6) (6 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 12 x] + \frac{d}{d x} [2]) + (6 x^{2} - 12 x + 2) \frac{d}{d x} [6 x - 6]$

Step 2.2.3

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

$(6 x - 6) (6 (2 x) + \frac{d}{d x} [- 12 x] + \frac{d}{d x} [2]) + (6 x^{2} - 12 x + 2) \frac{d}{d x} [6 x - 6]$

Step 2.2.4

Multiply $2$ by $6$ .

$(6 x - 6) (12 x + \frac{d}{d x} [- 12 x] + \frac{d}{d x} [2]) + (6 x^{2} - 12 x + 2) \frac{d}{d x} [6 x - 6]$

Step 2.2.5

Since $- 12$ is constant with respect to $x$ , the derivative of $- 12 x$ with respect to $x$ is $- 12 \frac{d}{d x} [x]$ .

$(6 x - 6) (12 x - 12 \frac{d}{d x} [x] + \frac{d}{d x} [2]) + (6 x^{2} - 12 x + 2) \frac{d}{d x} [6 x - 6]$

Step 2.2.6

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

$(6 x - 6) (12 x - 12 \cdot 1 + \frac{d}{d x} [2]) + (6 x^{2} - 12 x + 2) \frac{d}{d x} [6 x - 6]$

Step 2.2.7

Multiply $- 12$ by $1$ .

$(6 x - 6) (12 x - 12 + \frac{d}{d x} [2]) + (6 x^{2} - 12 x + 2) \frac{d}{d x} [6 x - 6]$

Step 2.2.8

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$(6 x - 6) (12 x - 12 + 0) + (6 x^{2} - 12 x + 2) \frac{d}{d x} [6 x - 6]$

Step 2.2.9

Add $12 x - 12$ and $0$ .

$(6 x - 6) (12 x - 12) + (6 x^{2} - 12 x + 2) \frac{d}{d x} [6 x - 6]$

Step 2.2.10

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

$(6 x - 6) (12 x - 12) + (6 x^{2} - 12 x + 2) (\frac{d}{d x} [6 x] + \frac{d}{d x} [- 6])$

Step 2.2.11

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

$(6 x - 6) (12 x - 12) + (6 x^{2} - 12 x + 2) (6 \frac{d}{d x} [x] + \frac{d}{d x} [- 6])$

Step 2.2.12

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

$(6 x - 6) (12 x - 12) + (6 x^{2} - 12 x + 2) (6 \cdot 1 + \frac{d}{d x} [- 6])$

Step 2.2.13

Multiply $6$ by $1$ .

$(6 x - 6) (12 x - 12) + (6 x^{2} - 12 x + 2) (6 + \frac{d}{d x} [- 6])$

Step 2.2.14

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6$ with respect to $x$ is $0$ .

$(6 x - 6) (12 x - 12) + (6 x^{2} - 12 x + 2) (6 + 0)$

Step 2.2.15

Simplify the expression.

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Step 2.2.15.1

Add $6$ and $0$ .

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

Step 2.2.15.2

Move $6$ to the left of $6 x^{2} - 12 x + 2$ .

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

Step 2.3

Simplify.

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Step 2.3.1

Apply the distributive property.

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

Step 2.3.2

Apply the distributive property.

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

Step 2.3.3

Apply the distributive property.

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

Step 2.3.4

Apply the distributive property.

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

Step 2.3.5

Combine terms.

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Step 2.3.5.1

Multiply $12$ by $6$ .

$72 x \cdot x - 6 (12 x) + 6 x \cdot - 12 - 6 \cdot - 12 + 6 (6 x^{2}) + 6 (- 12 x) + 6 \cdot 2$

Step 2.3.5.2

Raise $x$ to the power of $1$ .

$72 (x^{1} x) - 6 (12 x) + 6 x \cdot - 12 - 6 \cdot - 12 + 6 (6 x^{2}) + 6 (- 12 x) + 6 \cdot 2$

Step 2.3.5.3

Raise $x$ to the power of $1$ .

$72 (x^{1} x^{1}) - 6 (12 x) + 6 x \cdot - 12 - 6 \cdot - 12 + 6 (6 x^{2}) + 6 (- 12 x) + 6 \cdot 2$

Step 2.3.5.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$72 x^{1 + 1} - 6 (12 x) + 6 x \cdot - 12 - 6 \cdot - 12 + 6 (6 x^{2}) + 6 (- 12 x) + 6 \cdot 2$

Step 2.3.5.5

Add $1$ and $1$ .

$72 x^{2} - 6 (12 x) + 6 x \cdot - 12 - 6 \cdot - 12 + 6 (6 x^{2}) + 6 (- 12 x) + 6 \cdot 2$

Step 2.3.5.6

Multiply $12$ by $- 6$ .

$72 x^{2} - 72 x + 6 x \cdot - 12 - 6 \cdot - 12 + 6 (6 x^{2}) + 6 (- 12 x) + 6 \cdot 2$

Step 2.3.5.7

Multiply $- 12$ by $6$ .

$72 x^{2} - 72 x - 72 x - 6 \cdot - 12 + 6 (6 x^{2}) + 6 (- 12 x) + 6 \cdot 2$

Step 2.3.5.8

Multiply $- 6$ by $- 12$ .

$72 x^{2} - 72 x - 72 x + 72 + 6 (6 x^{2}) + 6 (- 12 x) + 6 \cdot 2$

Step 2.3.5.9

Subtract $72 x$ from $- 72 x$ .

$72 x^{2} - 144 x + 72 + 6 (6 x^{2}) + 6 (- 12 x) + 6 \cdot 2$

Step 2.3.5.10

Multiply $6$ by $6$ .

$72 x^{2} - 144 x + 72 + 36 x^{2} + 6 (- 12 x) + 6 \cdot 2$

Step 2.3.5.11

Multiply $- 12$ by $6$ .

$72 x^{2} - 144 x + 72 + 36 x^{2} - 72 x + 6 \cdot 2$

Step 2.3.5.12

Multiply $6$ by $2$ .

$72 x^{2} - 144 x + 72 + 36 x^{2} - 72 x + 12$

Step 2.3.5.13

Add $72 x^{2}$ and $36 x^{2}$ .

$108 x^{2} - 144 x + 72 - 72 x + 12$

Step 2.3.5.14

Subtract $72 x$ from $- 144 x$ .

$108 x^{2} - 216 x + 72 + 12$

Step 2.3.5.15

Add $72$ and $12$ .

$f'' (x) = 108 x^{2} - 216 x + 84$