Calculus Examples

${(x + 7)}^{2} + {(\sqrt{x})}^{2}$

Step 1

Rewrite ${(x + 7)}^{2}$ as $(x + 7) (x + 7)$ .

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

Step 2

Expand $(x + 7) (x + 7)$ using the FOIL Method.

Tap for more steps...

Step 2.1

Apply the distributive property.

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

Step 3

Simplify and combine like terms.

Tap for more steps...

Step 3.1

Simplify each term.

Tap for more steps...

Step 3.1.1

Multiply $x$ by $x$ .

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

Step 3.1.2

Move $7$ to the left of $x$ .

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

Step 3.1.3

Multiply $7$ by $7$ .

$\frac{d}{d x} [x^{2} + 7 x + 7 x + 49 + {(\sqrt{x})}^{2}]$

Step 3.2

Add $7 x$ and $7 x$ .

$\frac{d}{d x} [x^{2} + 14 x + 49 + {(\sqrt{x})}^{2}]$

Step 4

Differentiate.

Tap for more steps...

Step 4.1

By the Sum Rule, the derivative of $x^{2} + 14 x + 49 + {(\sqrt{x})}^{2}$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [14 x] + \frac{d}{d x} [49] + \frac{d}{d x} [{(\sqrt{x})}^{2}]$ .

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [14 x] + \frac{d}{d x} [49] + \frac{d}{d x} [{(\sqrt{x})}^{2}]$

Step 4.2

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

$2 x + \frac{d}{d x} [14 x] + \frac{d}{d x} [49] + \frac{d}{d x} [{(\sqrt{x})}^{2}]$

Step 5

Evaluate $\frac{d}{d x} [14 x]$ .

Tap for more steps...

Step 5.1

Since $14$ is constant with respect to $x$ , the derivative of $14 x$ with respect to $x$ is $14 \frac{d}{d x} [x]$ .

$2 x + 14 \frac{d}{d x} [x] + \frac{d}{d x} [49] + \frac{d}{d x} [{(\sqrt{x})}^{2}]$

Step 5.2

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

$2 x + 14 \cdot 1 + \frac{d}{d x} [49] + \frac{d}{d x} [{(\sqrt{x})}^{2}]$

Step 5.3

Multiply $14$ by $1$ .

$2 x + 14 + \frac{d}{d x} [49] + \frac{d}{d x} [{(\sqrt{x})}^{2}]$

Step 6

Since $49$ is constant with respect to $x$ , the derivative of $49$ with respect to $x$ is $0$ .

$2 x + 14 + 0 + \frac{d}{d x} [{(\sqrt{x})}^{2}]$

Step 7

Evaluate $\frac{d}{d x} [{(\sqrt{x})}^{2}]$ .

Tap for more steps...

Step 7.1

Rewrite ${(\sqrt{x})}^{2}$ as $x$ .

Tap for more steps...

Step 7.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$2 x + 14 + 0 + \frac{d}{d x} [{(x^{\frac{1}{2}})}^{2}]$

Step 7.1.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$2 x + 14 + 0 + \frac{d}{d x} [x^{\frac{1}{2} \cdot 2}]$

Step 7.1.3

Combine $\frac{1}{2}$ and $2$ .

$2 x + 14 + 0 + \frac{d}{d x} [x^{\frac{2}{2}}]$

Step 7.1.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 7.1.4.1

Cancel the common factor.

$2 x + 14 + 0 + \frac{d}{d x} [x^{\frac{2}{2}}]$

Step 7.1.4.2

Rewrite the expression.

$2 x + 14 + 0 + \frac{d}{d x} [x^{1}]$

Step 7.1.5

Simplify.

$2 x + 14 + 0 + \frac{d}{d x} [x]$

Step 7.2

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

$2 x + 14 + 0 + 1$

Step 8

Combine terms.

Tap for more steps...

Step 8.1

Add $2 x + 14$ and $0$ .

$2 x + 14 + 1$

Step 8.2

Add $14$ and $1$ .

$2 x + 15$