Calculus Examples

$q (x) = \sin (x) \sec (2 x)$

Step 1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = \sin (x)$ and $g (x) = \sec (2 x)$ .

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

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sec (x)$ and $g (x) = 2 x$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $2 x$ .

$\sin (x) (\frac{d}{d u} [\sec (u)] \frac{d}{d x} [2 x]) + \sec (2 x) \frac{d}{d x} [\sin (x)]$

Step 2.2

The derivative of $\sec (u)$ with respect to $u$ is $\sec (u) \tan (u)$ .

$\sin (x) (\sec (u) \tan (u) \frac{d}{d x} [2 x]) + \sec (2 x) \frac{d}{d x} [\sin (x)]$

Step 2.3

Replace all occurrences of $u$ with $2 x$ .

$\sin (x) (\sec (2 x) \tan (2 x) \frac{d}{d x} [2 x]) + \sec (2 x) \frac{d}{d x} [\sin (x)]$

Step 3

Differentiate.

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Step 3.1

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

$\sin (x) \sec (2 x) \tan (2 x) (2 \frac{d}{d x} [x]) + \sec (2 x) \frac{d}{d x} [\sin (x)]$

Step 3.2

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

$\sin (x) \sec (2 x) \tan (2 x) (2 \cdot 1) + \sec (2 x) \frac{d}{d x} [\sin (x)]$

Step 3.3

Simplify the expression.

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Step 3.3.1

Multiply $2$ by $1$ .

$\sin (x) \sec (2 x) \tan (2 x) \cdot 2 + \sec (2 x) \frac{d}{d x} [\sin (x)]$

Step 3.3.2

Move $2$ to the left of $\sin (x) \sec (2 x) \tan (2 x)$ .

$2 \cdot (\sin (x) \sec (2 x) \tan (2 x)) + \sec (2 x) \frac{d}{d x} [\sin (x)]$

Step 4

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

$2 \sin (x) \sec (2 x) \tan (2 x) + \sec (2 x) \cos (x)$

Step 5

Simplify.

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Step 5.1

Reorder terms.

$2 \sec (2 x) \sin (x) \tan (2 x) + \cos (x) \sec (2 x)$

Step 5.2

Simplify each term.

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Step 5.2.1

Rewrite $\sec (2 x)$ in terms of sines and cosines.

$2 \frac{1}{\cos (2 x)} \sin (x) \tan (2 x) + \cos (x) \sec (2 x)$

Step 5.2.2

Combine $2$ and $\frac{1}{\cos (2 x)}$ .

$\frac{2}{\cos (2 x)} \sin (x) \tan (2 x) + \cos (x) \sec (2 x)$

Step 5.2.3

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

$\frac{2 \sin (x)}{\cos (2 x)} \tan (2 x) + \cos (x) \sec (2 x)$

Step 5.2.4

Rewrite $\tan (2 x)$ in terms of sines and cosines.

$\frac{2 \sin (x)}{\cos (2 x)} \cdot \frac{\sin (2 x)}{\cos (2 x)} + \cos (x) \sec (2 x)$

Step 5.2.5

Multiply $\frac{2 \sin (x)}{\cos (2 x)} \cdot \frac{\sin (2 x)}{\cos (2 x)}$ .

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Step 5.2.5.1

Multiply $\frac{2 \sin (x)}{\cos (2 x)}$ by $\frac{\sin (2 x)}{\cos (2 x)}$ .

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

Step 5.2.5.2

Raise $\cos (2 x)$ to the power of $1$ .

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

Step 5.2.5.3

Raise $\cos (2 x)$ to the power of $1$ .

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

Step 5.2.5.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 5.2.5.5

Add $1$ and $1$ .

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

Step 5.2.6

Rewrite $\sec (2 x)$ in terms of sines and cosines.

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

Step 5.2.7

Combine $\cos (x)$ and $\frac{1}{\cos (2 x)}$ .

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

Step 5.3

Simplify each term.

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Step 5.3.1

Factor $\cos (2 x)$ out of $\cos^{2} (2 x)$ .

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

Step 5.3.2

Separate fractions.

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

Step 5.3.3

Rewrite $\frac{\sin (x)}{\cos (2 x)}$ as a product.

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

Step 5.3.4

Write $\sin (x)$ as a fraction with denominator $1$ .

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

Step 5.3.5

Simplify.

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Step 5.3.5.1

Divide $\sin (x)$ by $1$ .

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

Step 5.3.5.2

Convert from $\frac{1}{\cos (2 x)}$ to $\sec (2 x)$ .

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

Step 5.3.6

Separate fractions.

$\frac{2}{1} \cdot \frac{\sin (2 x)}{\cos (2 x)} (\sin (x) \sec (2 x)) + \frac{\cos (x)}{\cos (2 x)}$

Step 5.3.7

Convert from $\frac{\sin (2 x)}{\cos (2 x)}$ to $\tan (2 x)$ .

$\frac{2}{1} \tan (2 x) (\sin (x) \sec (2 x)) + \frac{\cos (x)}{\cos (2 x)}$

Step 5.3.8

Divide $2$ by $1$ .

$2 \tan (2 x) (\sin (x) \sec (2 x)) + \frac{\cos (x)}{\cos (2 x)}$

Step 5.3.9

Rewrite $\frac{\cos (x)}{\cos (2 x)}$ as a product.

$2 \tan (2 x) \sin (x) \sec (2 x) + \cos (x) \frac{1}{\cos (2 x)}$

Step 5.3.10

Write $\cos (x)$ as a fraction with denominator $1$ .

$2 \tan (2 x) \sin (x) \sec (2 x) + \frac{\cos (x)}{1} \cdot \frac{1}{\cos (2 x)}$

Step 5.3.11

Simplify.

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Step 5.3.11.1

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

$2 \tan (2 x) \sin (x) \sec (2 x) + \cos (x) \frac{1}{\cos (2 x)}$

Step 5.3.11.2

Convert from $\frac{1}{\cos (2 x)}$ to $\sec (2 x)$ .

$2 \tan (2 x) \sin (x) \sec (2 x) + \cos (x) \sec (2 x)$