Calculus Examples

$f (x) = 2 x^{2} - 5 x + 3$ , $a = 1$

Step 1

By the Sum Rule, the derivative of $2 x^{2} - 5 x + 3$ with respect to $x$ is $\frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [- 5 x] + \frac{d}{d x} [3]$ .

$\frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [- 5 x] + \frac{d}{d x} [3]$

Step 2

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

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Step 2.1

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

$2 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 5 x] + \frac{d}{d x} [3]$

Step 2.2

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

$2 (2 x) + \frac{d}{d x} [- 5 x] + \frac{d}{d x} [3]$

Step 2.3

Multiply $2$ by $2$ .

$4 x + \frac{d}{d x} [- 5 x] + \frac{d}{d x} [3]$

Step 3

Evaluate $\frac{d}{d x} [- 5 x]$ .

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Step 3.1

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5 x$ with respect to $x$ is $- 5 \frac{d}{d x} [x]$ .

$4 x - 5 \frac{d}{d x} [x] + \frac{d}{d x} [3]$

Step 3.2

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

$4 x - 5 \cdot 1 + \frac{d}{d x} [3]$

Step 3.3

Multiply $- 5$ by $1$ .

$4 x - 5 + \frac{d}{d x} [3]$

Step 4

Differentiate using the Constant Rule.

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Step 4.1

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$4 x - 5 + 0$

Step 4.2

Add $4 x - 5$ and $0$ .

$4 x - 5$