Calculus Examples

$f (x) = {(5 x + 6)}^{2}$

Step 1

This derivative could not be completed using the product rule. Mathway will use another method.

Step 2

Rewrite ${(5 x + 6)}^{2}$ as $(5 x + 6) (5 x + 6)$ .

$\frac{d}{d x} [(5 x + 6) (5 x + 6)]$

Step 3

Expand $(5 x + 6) (5 x + 6)$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

$\frac{d}{d x} [5 x (5 x + 6) + 6 (5 x + 6)]$

Step 3.2

Apply the distributive property.

$\frac{d}{d x} [5 x (5 x) + 5 x \cdot 6 + 6 (5 x + 6)]$

Step 3.3

Apply the distributive property.

$\frac{d}{d x} [5 x (5 x) + 5 x \cdot 6 + 6 (5 x) + 6 \cdot 6]$

Step 4

Simplify and combine like terms.

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Step 4.1

Simplify each term.

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Step 4.1.1

Rewrite using the commutative property of multiplication.

$\frac{d}{d x} [5 \cdot 5 x \cdot x + 5 x \cdot 6 + 6 (5 x) + 6 \cdot 6]$

Step 4.1.2

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

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Step 4.1.2.1

Move $x$ .

$\frac{d}{d x} [5 \cdot 5 (x \cdot x) + 5 x \cdot 6 + 6 (5 x) + 6 \cdot 6]$

Step 4.1.2.2

Multiply $x$ by $x$ .

$\frac{d}{d x} [5 \cdot 5 x^{2} + 5 x \cdot 6 + 6 (5 x) + 6 \cdot 6]$

Step 4.1.3

Multiply $5$ by $5$ .

$\frac{d}{d x} [25 x^{2} + 5 x \cdot 6 + 6 (5 x) + 6 \cdot 6]$

Step 4.1.4

Multiply $6$ by $5$ .

$\frac{d}{d x} [25 x^{2} + 30 x + 6 (5 x) + 6 \cdot 6]$

Step 4.1.5

Multiply $5$ by $6$ .

$\frac{d}{d x} [25 x^{2} + 30 x + 30 x + 6 \cdot 6]$

Step 4.1.6

Multiply $6$ by $6$ .

$\frac{d}{d x} [25 x^{2} + 30 x + 30 x + 36]$

Step 4.2

Add $30 x$ and $30 x$ .

$\frac{d}{d x} [25 x^{2} + 60 x + 36]$

Step 5

By the Sum Rule, the derivative of $25 x^{2} + 60 x + 36$ with respect to $x$ is $\frac{d}{d x} [25 x^{2}] + \frac{d}{d x} [60 x] + \frac{d}{d x} [36]$ .

$\frac{d}{d x} [25 x^{2}] + \frac{d}{d x} [60 x] + \frac{d}{d x} [36]$

Step 6

Since $25$ is constant with respect to $x$ , the derivative of $25 x^{2}$ with respect to $x$ is $25 \frac{d}{d x} [x^{2}]$ .

$25 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [60 x] + \frac{d}{d x} [36]$

Step 7

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

$25 (2 x) + \frac{d}{d x} [60 x] + \frac{d}{d x} [36]$

Step 8

Multiply $2$ by $25$ .

$50 x + \frac{d}{d x} [60 x] + \frac{d}{d x} [36]$

Step 9

Since $60$ is constant with respect to $x$ , the derivative of $60 x$ with respect to $x$ is $60 \frac{d}{d x} [x]$ .

$50 x + 60 \frac{d}{d x} [x] + \frac{d}{d x} [36]$

Step 10

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

$50 x + 60 \cdot 1 + \frac{d}{d x} [36]$

Step 11

Multiply $60$ by $1$ .

$50 x + 60 + \frac{d}{d x} [36]$

Step 12

Since $36$ is constant with respect to $x$ , the derivative of $36$ with respect to $x$ is $0$ .

$50 x + 60 + 0$

Step 13

Add $50 x + 60$ and $0$ .

$50 x + 60$