Calculus Examples

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

Step 1

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

$\frac{d}{d x} [\sin (2 x) \frac{1}{\cos (x)}]$

Step 2

Write $\sin (2 x)$ as a fraction with denominator $1$ .

$\frac{d}{d x} [\frac{\sin (2 x)}{1} \cdot \frac{1}{\cos (x)}]$

Step 3

Simplify.

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Step 3.1

Divide $\sin (2 x)$ by $1$ .

$\frac{d}{d x} [\sin (2 x) \frac{1}{\cos (x)}]$

Step 3.2

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

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

Step 4

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 $f (x) = \sin (2 x)$ and $g (x) = \sec (x)$ .

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

Step 5

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

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

Step 6

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

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Step 6.1

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

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

Step 6.2

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

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

Step 6.3

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

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

Step 7

Differentiate.

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Step 7.1

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

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

Step 7.2

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

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

Step 7.3

Simplify the expression.

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Step 7.3.1

Multiply $2$ by $1$ .

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

Step 7.3.2

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

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

Step 8

Simplify.

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Step 8.1

Reorder terms.

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

Step 8.2

Simplify each term.

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Step 8.2.1

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

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

Step 8.2.2

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

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

Step 8.2.3

Apply the sine double-angle identity.

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

Step 8.2.4

Cancel the common factor of $\cos (x)$ .

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Step 8.2.4.1

Cancel the common factor.

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

Step 8.2.4.2

Divide $2 \sin (x)$ by $1$ .

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

Step 8.2.5

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

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

Step 8.2.6

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

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Step 8.2.6.1

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

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

Step 8.2.6.2

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

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

Step 8.2.6.3

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

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

Step 8.2.6.4

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

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

Step 8.2.6.5

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

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

Step 8.2.6.6

Add $1$ and $1$ .

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

Step 8.2.7

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

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

Step 8.2.8

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

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Step 8.2.8.1

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

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

Step 8.2.8.2

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

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

Step 8.3

Simplify each term.

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Step 8.3.1

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

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

Step 8.3.2

Separate fractions.

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

Step 8.3.3

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

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

Step 8.3.4

Divide $2 (\sin (x))$ by $1$ .

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

Step 8.3.5

Separate fractions.

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

Step 8.3.6

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

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

Step 8.3.7

Write $\cos (2 x)$ as a fraction with denominator $1$ .

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

Step 8.3.8

Simplify.

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Step 8.3.8.1

Divide $\cos (2 x)$ by $1$ .

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

Step 8.3.8.2

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

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

Step 8.3.9

Divide $2$ by $1$ .

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