Calculus Examples

$4 x^{2} - x + 5$

Step 1

By the Sum Rule, the derivative of $4 x^{2} - x + 5$ with respect to $x$ is $\frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [- x] + \frac{d}{d x} [5]$ .

$\frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [- x] + \frac{d}{d x} [5]$

Step 2

Evaluate $\frac{d}{d x} [4 x^{2}]$ .

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Step 2.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{2}$ with respect to $x$ is $4 \frac{d}{d x} [x^{2}]$ .

$4 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- x] + \frac{d}{d x} [5]$

Step 2.2

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

$4 (2 x) + \frac{d}{d x} [- x] + \frac{d}{d x} [5]$

Step 2.3

Multiply $2$ by $4$ .

$8 x + \frac{d}{d x} [- x] + \frac{d}{d x} [5]$

Step 3

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

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Step 3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

$8 x - \frac{d}{d x} [x] + \frac{d}{d x} [5]$

Step 3.2

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

$8 x - 1 \cdot 1 + \frac{d}{d x} [5]$

Step 3.3

Multiply $- 1$ by $1$ .

$8 x - 1 + \frac{d}{d x} [5]$

Step 4

Differentiate using the Constant Rule.

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Step 4.1

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$8 x - 1 + 0$

Step 4.2

Add $8 x - 1$ and $0$ .

$8 x - 1$