Calculus Examples

$y = 4 x^{3} + 2 x^{2} + 6 x - 8 + \frac{1}{x}$

Step 1

By the Sum Rule, the derivative of $4 x^{3} + 2 x^{2} + 6 x - 8 + \frac{1}{x}$ with respect to $x$ is $\frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [6 x] + \frac{d}{d x} [- 8] + \frac{d}{d x} [\frac{1}{x}]$ .

$\frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [6 x] + \frac{d}{d x} [- 8] + \frac{d}{d x} [\frac{1}{x}]$

Step 2

Evaluate $\frac{d}{d x} [4 x^{3}]$ .

Tap for more steps...

Step 2.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{3}$ with respect to $x$ is $4 \frac{d}{d x} [x^{3}]$ .

$4 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [6 x] + \frac{d}{d x} [- 8] + \frac{d}{d x} [\frac{1}{x}]$

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$4 (3 x^{2}) + \frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [6 x] + \frac{d}{d x} [- 8] + \frac{d}{d x} [\frac{1}{x}]$

Step 2.3

Multiply $3$ by $4$ .

$12 x^{2} + \frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [6 x] + \frac{d}{d x} [- 8] + \frac{d}{d x} [\frac{1}{x}]$

Step 3

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

Tap for more steps...

Step 3.1

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

$12 x^{2} + 2 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [6 x] + \frac{d}{d x} [- 8] + \frac{d}{d x} [\frac{1}{x}]$

Step 3.2

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

$12 x^{2} + 2 (2 x) + \frac{d}{d x} [6 x] + \frac{d}{d x} [- 8] + \frac{d}{d x} [\frac{1}{x}]$

Step 3.3

Multiply $2$ by $2$ .

$12 x^{2} + 4 x + \frac{d}{d x} [6 x] + \frac{d}{d x} [- 8] + \frac{d}{d x} [\frac{1}{x}]$

Step 4

Evaluate $\frac{d}{d x} [6 x]$ .

Tap for more steps...

Step 4.1

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

$12 x^{2} + 4 x + 6 \frac{d}{d x} [x] + \frac{d}{d x} [- 8] + \frac{d}{d x} [\frac{1}{x}]$

Step 4.2

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

$12 x^{2} + 4 x + 6 \cdot 1 + \frac{d}{d x} [- 8] + \frac{d}{d x} [\frac{1}{x}]$

Step 4.3

Multiply $6$ by $1$ .

$12 x^{2} + 4 x + 6 + \frac{d}{d x} [- 8] + \frac{d}{d x} [\frac{1}{x}]$

Step 5

Since $- 8$ is constant with respect to $x$ , the derivative of $- 8$ with respect to $x$ is $0$ .

$12 x^{2} + 4 x + 6 + 0 + \frac{d}{d x} [\frac{1}{x}]$

Step 6

Evaluate $\frac{d}{d x} [\frac{1}{x}]$ .

Tap for more steps...

Step 6.1

Rewrite $\frac{1}{x}$ as $x^{- 1}$ .

$12 x^{2} + 4 x + 6 + 0 + \frac{d}{d x} [x^{- 1}]$

Step 6.2

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

$12 x^{2} + 4 x + 6 + 0 - x^{- 2}$

Step 7

Simplify.

Tap for more steps...

Step 7.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$12 x^{2} + 4 x + 6 + 0 - \frac{1}{x^{2}}$

Step 7.2

Add $12 x^{2} + 4 x + 6$ and $0$ .

$12 x^{2} + 4 x + 6 - \frac{1}{x^{2}}$

Step 7.3

Reorder terms.

$12 x^{2} + 4 x - \frac{1}{x^{2}} + 6$