Calculus Examples

$y = 2 x^{2} + 5$

Step 1

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

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

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

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

Step 2.2

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

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

Step 2.3

Multiply $2$ by $2$ .

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

Step 3

Differentiate using the Constant Rule.

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Step 3.1

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

$4 x + 0$

Step 3.2

Add $4 x$ and $0$ .

$4 x$