Calculus Examples

$r (a) = {(3 a + 1)}^{2}$

Step 1

Rewrite ${(3 a + 1)}^{2}$ as $(3 a + 1) (3 a + 1)$ .

$\frac{d}{d a} [(3 a + 1) (3 a + 1)]$

Step 2

Expand $(3 a + 1) (3 a + 1)$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$\frac{d}{d a} [3 a (3 a + 1) + 1 (3 a + 1)]$

Step 2.2

Apply the distributive property.

$\frac{d}{d a} [3 a (3 a) + 3 a \cdot 1 + 1 (3 a + 1)]$

Step 2.3

Apply the distributive property.

$\frac{d}{d a} [3 a (3 a) + 3 a \cdot 1 + 1 (3 a) + 1 \cdot 1]$

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Rewrite using the commutative property of multiplication.

$\frac{d}{d a} [3 \cdot 3 a \cdot a + 3 a \cdot 1 + 1 (3 a) + 1 \cdot 1]$

Step 3.1.2

Multiply $a$ by $a$ by adding the exponents.

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Step 3.1.2.1

Move $a$ .

$\frac{d}{d a} [3 \cdot 3 (a \cdot a) + 3 a \cdot 1 + 1 (3 a) + 1 \cdot 1]$

Step 3.1.2.2

Multiply $a$ by $a$ .

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

Step 3.1.3

Multiply $3$ by $3$ .

$\frac{d}{d a} [9 a^{2} + 3 a \cdot 1 + 1 (3 a) + 1 \cdot 1]$

Step 3.1.4

Multiply $3$ by $1$ .

$\frac{d}{d a} [9 a^{2} + 3 a + 1 (3 a) + 1 \cdot 1]$

Step 3.1.5

Multiply $3 a$ by $1$ .

$\frac{d}{d a} [9 a^{2} + 3 a + 3 a + 1 \cdot 1]$

Step 3.1.6

Multiply $1$ by $1$ .

$\frac{d}{d a} [9 a^{2} + 3 a + 3 a + 1]$

Step 3.2

Add $3 a$ and $3 a$ .

$\frac{d}{d a} [9 a^{2} + 6 a + 1]$

Step 4

By the Sum Rule, the derivative of $9 a^{2} + 6 a + 1$ with respect to $a$ is $\frac{d}{d a} [9 a^{2}] + \frac{d}{d a} [6 a] + \frac{d}{d a} [1]$ .

$\frac{d}{d a} [9 a^{2}] + \frac{d}{d a} [6 a] + \frac{d}{d a} [1]$

Step 5

Since $9$ is constant with respect to $a$ , the derivative of $9 a^{2}$ with respect to $a$ is $9 \frac{d}{d a} [a^{2}]$ .

$9 \frac{d}{d a} [a^{2}] + \frac{d}{d a} [6 a] + \frac{d}{d a} [1]$

Step 6

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

$9 (2 a) + \frac{d}{d a} [6 a] + \frac{d}{d a} [1]$

Step 7

Multiply $2$ by $9$ .

$18 a + \frac{d}{d a} [6 a] + \frac{d}{d a} [1]$

Step 8

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

$18 a + 6 \frac{d}{d a} [a] + \frac{d}{d a} [1]$

Step 9

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

$18 a + 6 \cdot 1 + \frac{d}{d a} [1]$

Step 10

Multiply $6$ by $1$ .

$18 a + 6 + \frac{d}{d a} [1]$

Step 11

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

$18 a + 6 + 0$

Step 12

Add $18 a + 6$ and $0$ .

$18 a + 6$