Calculus Examples

$\frac{\sqrt{2 x^{2}}}{\cos (x)}$

Step 1

Rewrite $\frac{d}{d x} [\frac{\sqrt{2 x^{2}}}{\cos (x)}]$ as $\frac{d}{d x} [\frac{x \sqrt{2}}{\cos (x)}]$ .

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Step 1.1

Reorder $2$ and $x^{2}$ .

$\frac{d}{d x} [\frac{\sqrt{x^{2} \cdot 2}}{\cos (x)}]$

Step 1.2

Pull terms out from under the radical.

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

Step 2

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

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

Step 3

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = x$ and $g (x) = \cos (x)$ .

$\sqrt{2} \frac{\cos (x) \frac{d}{d x} [x] - x \frac{d}{d x} [\cos (x)]}{\cos^{2} (x)}$

Step 4

Differentiate using the Power Rule.

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Step 4.1

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

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

Step 4.2

Multiply $\cos (x)$ by $1$ .

$\sqrt{2} \frac{\cos (x) - x \frac{d}{d x} [\cos (x)]}{\cos^{2} (x)}$

Step 5

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

$\sqrt{2} \frac{\cos (x) - x (- \sin (x))}{\cos^{2} (x)}$

Step 6

Combine fractions.

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Step 6.1

Multiply $- 1$ by $- 1$ .

$\sqrt{2} \frac{\cos (x) + 1 x \sin (x)}{\cos^{2} (x)}$

Step 6.2

Multiply $x$ by $1$ .

$\sqrt{2} \frac{\cos (x) + x \sin (x)}{\cos^{2} (x)}$

Step 6.3

Combine $\sqrt{2}$ and $\frac{\cos (x) + x \sin (x)}{\cos^{2} (x)}$ .

$\frac{\sqrt{2} (\cos (x) + x \sin (x))}{\cos^{2} (x)}$

Step 7

Simplify.

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Step 7.1

Apply the distributive property.

$\frac{\sqrt{2} \cos (x) + \sqrt{2} (x \sin (x))}{\cos^{2} (x)}$

Step 7.2

Reorder terms.

$\frac{x \sqrt{2} \sin (x) + \sqrt{2} \cos (x)}{\cos^{2} (x)}$