Calculus Examples

$f (x) = {(x^{2} + 6 x - 7)}^{2}$

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 $g (x) = x^{2} + 6 x - 7$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $x^{2} + 6 x - 7$ .

$\frac{d}{d u} [u^{2}] \frac{d}{d x} [x^{2} + 6 x - 7]$

Step 1.2

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

$2 u \frac{d}{d x} [x^{2} + 6 x - 7]$

Step 1.3

Replace all occurrences of $u$ with $x^{2} + 6 x - 7$ .

$2 (x^{2} + 6 x - 7) \frac{d}{d x} [x^{2} + 6 x - 7]$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $x^{2} + 6 x - 7$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [6 x] + \frac{d}{d x} [- 7]$ .

$2 (x^{2} + 6 x - 7) (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [6 x] + \frac{d}{d x} [- 7])$

Step 2.2

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

$2 (x^{2} + 6 x - 7) (2 x + \frac{d}{d x} [6 x] + \frac{d}{d x} [- 7])$

Step 2.3

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

$2 (x^{2} + 6 x - 7) (2 x + 6 \frac{d}{d x} [x] + \frac{d}{d x} [- 7])$

Step 2.4

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

$2 (x^{2} + 6 x - 7) (2 x + 6 \cdot 1 + \frac{d}{d x} [- 7])$

Step 2.5

Multiply $6$ by $1$ .

$2 (x^{2} + 6 x - 7) (2 x + 6 + \frac{d}{d x} [- 7])$

Step 2.6

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

$2 (x^{2} + 6 x - 7) (2 x + 6 + 0)$

Step 2.7

Add $2 x + 6$ and $0$ .

$2 (x^{2} + 6 x - 7) (2 x + 6)$

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$(2 x^{2} + 2 (6 x) + 2 \cdot - 7) (2 x + 6)$

Step 3.2

Combine terms.

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Step 3.2.1

Multiply $6$ by $2$ .

$(2 x^{2} + 12 x + 2 \cdot - 7) (2 x + 6)$

Step 3.2.2

Multiply $2$ by $- 7$ .

$(2 x^{2} + 12 x - 14) (2 x + 6)$

Step 3.3

Reorder the factors of $(2 x^{2} + 12 x - 14) (2 x + 6)$ .

$(2 x + 6) (2 x^{2} + 12 x - 14)$

Step 3.4

Expand $(2 x + 6) (2 x^{2} + 12 x - 14)$ by multiplying each term in the first expression by each term in the second expression.

$2 x (2 x^{2}) + 2 x (12 x) + 2 x \cdot - 14 + 6 (2 x^{2}) + 6 (12 x) + 6 \cdot - 14$

Step 3.5

Simplify each term.

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Step 3.5.1

Rewrite using the commutative property of multiplication.

$2 \cdot 2 x \cdot x^{2} + 2 x (12 x) + 2 x \cdot - 14 + 6 (2 x^{2}) + 6 (12 x) + 6 \cdot - 14$

Step 3.5.2

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 3.5.2.1

Move $x^{2}$ .

$2 \cdot 2 (x^{2} x) + 2 x (12 x) + 2 x \cdot - 14 + 6 (2 x^{2}) + 6 (12 x) + 6 \cdot - 14$

Step 3.5.2.2

Multiply $x^{2}$ by $x$ .

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Step 3.5.2.2.1

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

$2 \cdot 2 (x^{2} x^{1}) + 2 x (12 x) + 2 x \cdot - 14 + 6 (2 x^{2}) + 6 (12 x) + 6 \cdot - 14$

Step 3.5.2.2.2

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

$2 \cdot 2 x^{2 + 1} + 2 x (12 x) + 2 x \cdot - 14 + 6 (2 x^{2}) + 6 (12 x) + 6 \cdot - 14$

Step 3.5.2.3

Add $2$ and $1$ .

$2 \cdot 2 x^{3} + 2 x (12 x) + 2 x \cdot - 14 + 6 (2 x^{2}) + 6 (12 x) + 6 \cdot - 14$

Step 3.5.3

Multiply $2$ by $2$ .

$4 x^{3} + 2 x (12 x) + 2 x \cdot - 14 + 6 (2 x^{2}) + 6 (12 x) + 6 \cdot - 14$

Step 3.5.4

Rewrite using the commutative property of multiplication.

$4 x^{3} + 2 \cdot 12 x \cdot x + 2 x \cdot - 14 + 6 (2 x^{2}) + 6 (12 x) + 6 \cdot - 14$

Step 3.5.5

Multiply $x$ by $x$ by adding the exponents.

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Step 3.5.5.1

Move $x$ .

$4 x^{3} + 2 \cdot 12 (x \cdot x) + 2 x \cdot - 14 + 6 (2 x^{2}) + 6 (12 x) + 6 \cdot - 14$

Step 3.5.5.2

Multiply $x$ by $x$ .

$4 x^{3} + 2 \cdot 12 x^{2} + 2 x \cdot - 14 + 6 (2 x^{2}) + 6 (12 x) + 6 \cdot - 14$

Step 3.5.6

Multiply $2$ by $12$ .

$4 x^{3} + 24 x^{2} + 2 x \cdot - 14 + 6 (2 x^{2}) + 6 (12 x) + 6 \cdot - 14$

Step 3.5.7

Multiply $- 14$ by $2$ .

$4 x^{3} + 24 x^{2} - 28 x + 6 (2 x^{2}) + 6 (12 x) + 6 \cdot - 14$

Step 3.5.8

Multiply $2$ by $6$ .

$4 x^{3} + 24 x^{2} - 28 x + 12 x^{2} + 6 (12 x) + 6 \cdot - 14$

Step 3.5.9

Multiply $12$ by $6$ .

$4 x^{3} + 24 x^{2} - 28 x + 12 x^{2} + 72 x + 6 \cdot - 14$

Step 3.5.10

Multiply $6$ by $- 14$ .

$4 x^{3} + 24 x^{2} - 28 x + 12 x^{2} + 72 x - 84$

Step 3.6

Add $24 x^{2}$ and $12 x^{2}$ .

$4 x^{3} + 36 x^{2} - 28 x + 72 x - 84$

Step 3.7

Add $- 28 x$ and $72 x$ .

$4 x^{3} + 36 x^{2} + 44 x - 84$