Calculus Examples

$y = \sqrt{3} x + 2 \cos (x)$

Step 1

By the Sum Rule, the derivative of $\sqrt{3} x + 2 \cos (x)$ with respect to $x$ is $\frac{d}{d x} [\sqrt{3} x] + \frac{d}{d x} [2 \cos (x)]$ .

$\frac{d}{d x} [\sqrt{3} x] + \frac{d}{d x} [2 \cos (x)]$

Step 2

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

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Step 2.1

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

$\sqrt{3} \frac{d}{d x} [x] + \frac{d}{d x} [2 \cos (x)]$

Step 2.2

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

$\sqrt{3} \cdot 1 + \frac{d}{d x} [2 \cos (x)]$

Step 2.3

Multiply $\sqrt{3}$ by $1$ .

$\sqrt{3} + \frac{d}{d x} [2 \cos (x)]$

Step 3

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

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Step 3.1

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

$\sqrt{3} + 2 \frac{d}{d x} [\cos (x)]$

Step 3.2

The derivative of $\cos (x)$ with respect to $x$ is $- \sin (x)$ .

$\sqrt{3} + 2 (- \sin (x))$

Step 3.3

Multiply $- 1$ by $2$ .

$\sqrt{3} - 2 \sin (x)$

Step 4

Reorder terms.

$- 2 \sin (x) + \sqrt{3}$