Calculus Examples

$j (x) = \frac{\sin^{5} (2 x)}{\cos^{5} (2 x)}$

Step 1

Let $x = f (x)$ , take the natural logarithm of both sides $\ln (x) = \ln (f (x))$ .

$\ln (j (x)) = \ln (\frac{\sin^{5} (2 x)}{\cos^{5} (2 x)})$

Step 2

Expand the right hand side.

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Step 2.1

Rewrite $\ln (\frac{\sin^{5} (2 x)}{\cos^{5} (2 x)})$ as $\ln (\sin^{5} (2 x)) - \ln (\cos^{5} (2 x))$ .

$\ln (j (x)) = \ln (\sin^{5} (2 x)) - \ln (\cos^{5} (2 x))$

Step 2.2

Expand $\ln (\sin^{5} (2 x))$ by moving $5$ outside the logarithm.

$\ln (j (x)) = 5 \ln (\sin (2 x)) - \ln (\cos^{5} (2 x))$

Step 2.3

Expand $\ln (\cos^{5} (2 x))$ by moving $5$ outside the logarithm.

$\ln (j (x)) = 5 \ln (\sin (2 x)) - (5 \ln (\cos (2 x)))$

Step 2.4

Multiply $5$ by $- 1$ .

$\ln (j (x)) = 5 \ln (\sin (2 x)) - 5 \ln (\cos (2 x))$

Step 3

Differentiate the expression using the chain rule, keeping in mind that $y$ is a function of $x$ .

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Step 3.1

Differentiate the left hand side $\ln (j (x))$ using the chain rule.

$\frac{x'}{x} = 5 \ln (\sin (2 x)) - 5 \ln (\cos (2 x))$

Step 3.2

Differentiate the right hand side.

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Step 3.2.1

Differentiate $5 \ln (\sin (2 x)) - 5 \ln (\cos (2 x))$ .

$\frac{x'}{x} = \frac{d}{d x} [5 \ln (\sin (2 x)) - 5 \ln (\cos (2 x))]$

Step 3.2.2

By the Sum Rule, the derivative of $5 \ln (\sin (2 x)) - 5 \ln (\cos (2 x))$ with respect to $x$ is $\frac{d}{d x} [5 \ln (\sin (2 x))] + \frac{d}{d x} [- 5 \ln (\cos (2 x))]$ .

$\frac{x'}{x} = \frac{d}{d x} [5 \ln (\sin (2 x))] + \frac{d}{d x} [- 5 \ln (\cos (2 x))]$

Step 3.2.3

Evaluate $\frac{d}{d x} [5 \ln (\sin (2 x))]$ .

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Step 3.2.3.1

Since $5$ is constant with respect to $x$ , the derivative of $5 \ln (\sin (2 x))$ with respect to $x$ is $5 \frac{d}{d x} [\ln (\sin (2 x))]$ .

$\frac{x'}{x} = 5 \frac{d}{d x} [\ln (\sin (2 x))] + \frac{d}{d x} [- 5 \ln (\cos (2 x))]$

Step 3.2.3.2

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

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Step 3.2.3.2.1

To apply the Chain Rule, set $u_{1}$ as $\sin (2 x)$ .

$\frac{x'}{x} = 5 (\frac{d}{d u_{1}} [\ln (u_{1})] \frac{d}{d x} [\sin (2 x)]) + \frac{d}{d x} [- 5 \ln (\cos (2 x))]$

Step 3.2.3.2.2

The derivative of $\ln (u_{1})$ with respect to $u_{1}$ is $\frac{1}{u_{1}}$ .

$\frac{x'}{x} = 5 (\frac{1}{u_{1}} \frac{d}{d x} [\sin (2 x)]) + \frac{d}{d x} [- 5 \ln (\cos (2 x))]$

Step 3.2.3.2.3

Replace all occurrences of $u_{1}$ with $\sin (2 x)$ .

$\frac{x'}{x} = 5 (\frac{1}{\sin (2 x)} \frac{d}{d x} [\sin (2 x)]) + \frac{d}{d x} [- 5 \ln (\cos (2 x))]$

Step 3.2.3.3

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 3.2.3.3.1

To apply the Chain Rule, set $u_{2}$ as $2 x$ .

$\frac{x'}{x} = 5 (\frac{1}{\sin (2 x)} (\frac{d}{d u_{2}} [\sin (u_{2})] \frac{d}{d x} [2 x])) + \frac{d}{d x} [- 5 \ln (\cos (2 x))]$

Step 3.2.3.3.2

The derivative of $\sin (u_{2})$ with respect to $u_{2}$ is $\cos (u_{2})$ .

$\frac{x'}{x} = 5 (\frac{1}{\sin (2 x)} (\cos (u_{2}) \frac{d}{d x} [2 x])) + \frac{d}{d x} [- 5 \ln (\cos (2 x))]$

Step 3.2.3.3.3

Replace all occurrences of $u_{2}$ with $2 x$ .

$\frac{x'}{x} = 5 (\frac{1}{\sin (2 x)} (\cos (2 x) \frac{d}{d x} [2 x])) + \frac{d}{d x} [- 5 \ln (\cos (2 x))]$

Step 3.2.3.4

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

$\frac{x'}{x} = 5 (\frac{1}{\sin (2 x)} (\cos (2 x) (2 \frac{d}{d x} [x]))) + \frac{d}{d x} [- 5 \ln (\cos (2 x))]$

Step 3.2.3.5

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

$\frac{x'}{x} = 5 (\frac{1}{\sin (2 x)} (\cos (2 x) (2 \cdot 1))) + \frac{d}{d x} [- 5 \ln (\cos (2 x))]$

Step 3.2.3.6

Convert from $\frac{1}{\sin (2 x)}$ to $\csc (2 x)$ .

$\frac{x'}{x} = 5 (\csc (2 x) (\cos (2 x) (2 \cdot 1))) + \frac{d}{d x} [- 5 \ln (\cos (2 x))]$

Step 3.2.3.7

Multiply $2$ by $1$ .

$\frac{x'}{x} = 5 (\csc (2 x) (\cos (2 x) \cdot 2)) + \frac{d}{d x} [- 5 \ln (\cos (2 x))]$

Step 3.2.3.8

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

$\frac{x'}{x} = 5 (\csc (2 x) (2 \cdot \cos (2 x))) + \frac{d}{d x} [- 5 \ln (\cos (2 x))]$

Step 3.2.3.9

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

$\frac{x'}{x} = 5 (2 \cdot \csc (2 x) \cos (2 x)) + \frac{d}{d x} [- 5 \ln (\cos (2 x))]$

Step 3.2.3.10

Multiply $2$ by $5$ .

$\frac{x'}{x} = 10 \csc (2 x) \cos (2 x) + \frac{d}{d x} [- 5 \ln (\cos (2 x))]$

Step 3.2.4

Evaluate $\frac{d}{d x} [- 5 \ln (\cos (2 x))]$ .

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Step 3.2.4.1

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5 \ln (\cos (2 x))$ with respect to $x$ is $- 5 \frac{d}{d x} [\ln (\cos (2 x))]$ .

$\frac{x'}{x} = 10 \csc (2 x) \cos (2 x) - 5 \frac{d}{d x} [\ln (\cos (2 x))]$

Step 3.2.4.2

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

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Step 3.2.4.2.1

To apply the Chain Rule, set $u_{3}$ as $\cos (2 x)$ .

$\frac{x'}{x} = 10 \csc (2 x) \cos (2 x) - 5 (\frac{d}{d u_{3}} [\ln (u_{3})] \frac{d}{d x} [\cos (2 x)])$

Step 3.2.4.2.2

The derivative of $\ln (u_{3})$ with respect to $u_{3}$ is $\frac{1}{u_{3}}$ .

$\frac{x'}{x} = 10 \csc (2 x) \cos (2 x) - 5 (\frac{1}{u_{3}} \frac{d}{d x} [\cos (2 x)])$

Step 3.2.4.2.3

Replace all occurrences of $u_{3}$ with $\cos (2 x)$ .

$\frac{x'}{x} = 10 \csc (2 x) \cos (2 x) - 5 (\frac{1}{\cos (2 x)} \frac{d}{d x} [\cos (2 x)])$

Step 3.2.4.3

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

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Step 3.2.4.3.1

To apply the Chain Rule, set $u_{4}$ as $2 x$ .

$\frac{x'}{x} = 10 \csc (2 x) \cos (2 x) - 5 (\frac{1}{\cos (2 x)} (\frac{d}{d u_{4}} [\cos (u_{4})] \frac{d}{d x} [2 x]))$

Step 3.2.4.3.2

The derivative of $\cos (u_{4})$ with respect to $u_{4}$ is $- \sin (u_{4})$ .

$\frac{x'}{x} = 10 \csc (2 x) \cos (2 x) - 5 (\frac{1}{\cos (2 x)} (- \sin (u_{4}) \frac{d}{d x} [2 x]))$

Step 3.2.4.3.3

Replace all occurrences of $u_{4}$ with $2 x$ .

$\frac{x'}{x} = 10 \csc (2 x) \cos (2 x) - 5 (\frac{1}{\cos (2 x)} (- \sin (2 x) \frac{d}{d x} [2 x]))$

Step 3.2.4.4

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

$\frac{x'}{x} = 10 \csc (2 x) \cos (2 x) - 5 (\frac{1}{\cos (2 x)} (- \sin (2 x) (2 \frac{d}{d x} [x])))$

Step 3.2.4.5

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

$\frac{x'}{x} = 10 \csc (2 x) \cos (2 x) - 5 (\frac{1}{\cos (2 x)} (- \sin (2 x) (2 \cdot 1)))$

Step 3.2.4.6

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

$\frac{x'}{x} = 10 \csc (2 x) \cos (2 x) - 5 (\sec (2 x) (- \sin (2 x) (2 \cdot 1)))$

Step 3.2.4.7

Multiply $2$ by $1$ .

$\frac{x'}{x} = 10 \csc (2 x) \cos (2 x) - 5 (\sec (2 x) (- \sin (2 x) \cdot 2))$

Step 3.2.4.8

Multiply $2$ by $- 1$ .

$\frac{x'}{x} = 10 \csc (2 x) \cos (2 x) - 5 (\sec (2 x) (- 2 \sin (2 x)))$

Step 3.2.4.9

Multiply $- 2$ by $- 5$ .

$\frac{x'}{x} = 10 \csc (2 x) \cos (2 x) + 10 \sec (2 x) \sin (2 x)$

Step 3.2.5

Simplify.

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Step 3.2.5.1

Reorder terms.

$\frac{x'}{x} = 10 \cos (2 x) \csc (2 x) + 10 \sec (2 x) \sin (2 x)$

Step 3.2.5.2

Simplify each term.

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Step 3.2.5.2.1

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

$\frac{x'}{x} = 10 \cos (2 x) \frac{1}{\sin (2 x)} + 10 \sec (2 x) \sin (2 x)$

Step 3.2.5.2.2

Multiply $10 \cos (2 x) \frac{1}{\sin (2 x)}$ .

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Step 3.2.5.2.2.1

Combine $\frac{1}{\sin (2 x)}$ and $10$ .

$\frac{x'}{x} = \frac{10}{\sin (2 x)} \cos (2 x) + 10 \sec (2 x) \sin (2 x)$

Step 3.2.5.2.2.2

Combine $\frac{10}{\sin (2 x)}$ and $\cos (2 x)$ .

$\frac{x'}{x} = \frac{10 \cos (2 x)}{\sin (2 x)} + 10 \sec (2 x) \sin (2 x)$

Step 3.2.5.2.3

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

$\frac{x'}{x} = \frac{10 \cos (2 x)}{\sin (2 x)} + 10 \frac{1}{\cos (2 x)} \sin (2 x)$

Step 3.2.5.2.4

Combine $10$ and $\frac{1}{\cos (2 x)}$ .

$\frac{x'}{x} = \frac{10 \cos (2 x)}{\sin (2 x)} + \frac{10}{\cos (2 x)} \sin (2 x)$

Step 3.2.5.2.5

Combine $\frac{10}{\cos (2 x)}$ and $\sin (2 x)$ .

$\frac{x'}{x} = \frac{10 \cos (2 x)}{\sin (2 x)} + \frac{10 \sin (2 x)}{\cos (2 x)}$

Step 3.2.5.3

Simplify each term.

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Step 3.2.5.3.1

Separate fractions.

$\frac{x'}{x} = \frac{10}{1} \cdot \frac{\cos (2 x)}{\sin (2 x)} + \frac{10 \sin (2 x)}{\cos (2 x)}$

Step 3.2.5.3.2

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

$\frac{x'}{x} = \frac{10}{1} \cot (2 x) + \frac{10 \sin (2 x)}{\cos (2 x)}$

Step 3.2.5.3.3

Divide $10$ by $1$ .

$\frac{x'}{x} = 10 \cot (2 x) + \frac{10 \sin (2 x)}{\cos (2 x)}$

Step 3.2.5.3.4

Separate fractions.

$\frac{x'}{x} = 10 \cot (2 x) + \frac{10}{1} \cdot \frac{\sin (2 x)}{\cos (2 x)}$

Step 3.2.5.3.5

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

$\frac{x'}{x} = 10 \cot (2 x) + \frac{10}{1} \tan (2 x)$

Step 3.2.5.3.6

Divide $10$ by $1$ .

$\frac{x'}{x} = 10 \cot (2 x) + 10 \tan (2 x)$

Step 4

Isolate $x'$ and substitute the original function for $x$ in the right hand side.

$x' = (10 \cot (2 x) + 10 \tan (2 x)) \frac{\sin^{5} (2 x)}{\cos^{5} (2 x)}$

Step 5

Simplify the right hand side.

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Step 5.1

Simplify each term.

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Step 5.1.1

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

$x' = (10 \frac{\cos (2 x)}{\sin (2 x)} + 10 \tan (2 x)) \frac{\sin^{5} (2 x)}{\cos^{5} (2 x)}$

Step 5.1.2

Combine $10$ and $\frac{\cos (2 x)}{\sin (2 x)}$ .

$x' = (\frac{10 \cos (2 x)}{\sin (2 x)} + 10 \tan (2 x)) \frac{\sin^{5} (2 x)}{\cos^{5} (2 x)}$

Step 5.1.3

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

$x' = (\frac{10 \cos (2 x)}{\sin (2 x)} + 10 \frac{\sin (2 x)}{\cos (2 x)}) \frac{\sin^{5} (2 x)}{\cos^{5} (2 x)}$

Step 5.1.4

Combine $10$ and $\frac{\sin (2 x)}{\cos (2 x)}$ .

$x' = (\frac{10 \cos (2 x)}{\sin (2 x)} + \frac{10 \sin (2 x)}{\cos (2 x)}) \frac{\sin^{5} (2 x)}{\cos^{5} (2 x)}$

Step 5.2

Apply the distributive property.

$x' = \frac{10 \cos (2 x)}{\sin (2 x)} \cdot \frac{\sin^{5} (2 x)}{\cos^{5} (2 x)} + \frac{10 \sin (2 x)}{\cos (2 x)} \cdot \frac{\sin^{5} (2 x)}{\cos^{5} (2 x)}$

Step 5.3

Combine.

$x' = \frac{10 \cos (2 x) \sin^{5} (2 x)}{\sin (2 x) \cos^{5} (2 x)} + \frac{10 \sin (2 x)}{\cos (2 x)} \cdot \frac{\sin^{5} (2 x)}{\cos^{5} (2 x)}$

Step 5.4

Combine.

$x' = \frac{10 \cos (2 x) \sin^{5} (2 x)}{\sin (2 x) \cos^{5} (2 x)} + \frac{10 \sin (2 x) \sin^{5} (2 x)}{\cos (2 x) \cos^{5} (2 x)}$

Step 5.5

Simplify each term.

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Step 5.5.1

Cancel the common factor of $\cos (2 x)$ and $\cos^{5} (2 x)$ .

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Step 5.5.1.1

Factor $\cos (2 x)$ out of $10 \cos (2 x) \sin^{5} (2 x)$ .

$x' = \frac{\cos (2 x) (10 \sin^{5} (2 x))}{\sin (2 x) \cos^{5} (2 x)} + \frac{10 \sin (2 x) \sin^{5} (2 x)}{\cos (2 x) \cos^{5} (2 x)}$

Step 5.5.1.2

Cancel the common factors.

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Step 5.5.1.2.1

Factor $\cos (2 x)$ out of $\sin (2 x) \cos^{5} (2 x)$ .

$x' = \frac{\cos (2 x) (10 \sin^{5} (2 x))}{\cos (2 x) (\sin (2 x) \cos^{4} (2 x))} + \frac{10 \sin (2 x) \sin^{5} (2 x)}{\cos (2 x) \cos^{5} (2 x)}$

Step 5.5.1.2.2

Cancel the common factor.

$x' = \frac{\cos (2 x) (10 \sin^{5} (2 x))}{\cos (2 x) (\sin (2 x) \cos^{4} (2 x))} + \frac{10 \sin (2 x) \sin^{5} (2 x)}{\cos (2 x) \cos^{5} (2 x)}$

Step 5.5.1.2.3

Rewrite the expression.

$x' = \frac{10 \sin^{5} (2 x)}{\sin (2 x) \cos^{4} (2 x)} + \frac{10 \sin (2 x) \sin^{5} (2 x)}{\cos (2 x) \cos^{5} (2 x)}$

Step 5.5.2

Cancel the common factor of $\sin^{5} (2 x)$ and $\sin (2 x)$ .

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Step 5.5.2.1

Factor $\sin (2 x)$ out of $10 \sin^{5} (2 x)$ .

$x' = \frac{\sin (2 x) (10 \sin^{4} (2 x))}{\sin (2 x) \cos^{4} (2 x)} + \frac{10 \sin (2 x) \sin^{5} (2 x)}{\cos (2 x) \cos^{5} (2 x)}$

Step 5.5.2.2

Cancel the common factors.

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Step 5.5.2.2.1

Factor $\sin (2 x)$ out of $\sin (2 x) \cos^{4} (2 x)$ .

$x' = \frac{\sin (2 x) (10 \sin^{4} (2 x))}{\sin (2 x) (\cos^{4} (2 x))} + \frac{10 \sin (2 x) \sin^{5} (2 x)}{\cos (2 x) \cos^{5} (2 x)}$

Step 5.5.2.2.2

Cancel the common factor.

$x' = \frac{\sin (2 x) (10 \sin^{4} (2 x))}{\sin (2 x) \cos^{4} (2 x)} + \frac{10 \sin (2 x) \sin^{5} (2 x)}{\cos (2 x) \cos^{5} (2 x)}$

Step 5.5.2.2.3

Rewrite the expression.

$x' = \frac{10 \sin^{4} (2 x)}{\cos^{4} (2 x)} + \frac{10 \sin (2 x) \sin^{5} (2 x)}{\cos (2 x) \cos^{5} (2 x)}$

Step 5.5.3

Multiply $\sin (2 x)$ by $\sin^{5} (2 x)$ by adding the exponents.

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Step 5.5.3.1

Move $\sin^{5} (2 x)$ .

$x' = \frac{10 \sin^{4} (2 x)}{\cos^{4} (2 x)} + \frac{10 (\sin^{5} (2 x) \sin (2 x))}{\cos (2 x) \cos^{5} (2 x)}$

Step 5.5.3.2

Multiply $\sin^{5} (2 x)$ by $\sin (2 x)$ .

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Step 5.5.3.2.1

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

$x' = \frac{10 \sin^{4} (2 x)}{\cos^{4} (2 x)} + \frac{10 (\sin^{5} (2 x) \sin^{1} (2 x))}{\cos (2 x) \cos^{5} (2 x)}$

Step 5.5.3.2.2

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

$x' = \frac{10 \sin^{4} (2 x)}{\cos^{4} (2 x)} + \frac{10 {\sin (2 x)}^{5 + 1}}{\cos (2 x) \cos^{5} (2 x)}$

Step 5.5.3.3

Add $5$ and $1$ .

$x' = \frac{10 \sin^{4} (2 x)}{\cos^{4} (2 x)} + \frac{10 \sin^{6} (2 x)}{\cos (2 x) \cos^{5} (2 x)}$

Step 5.5.4

Multiply $\cos (2 x)$ by $\cos^{5} (2 x)$ by adding the exponents.

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Step 5.5.4.1

Multiply $\cos (2 x)$ by $\cos^{5} (2 x)$ .

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Step 5.5.4.1.1

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

$x' = \frac{10 \sin^{4} (2 x)}{\cos^{4} (2 x)} + \frac{10 \sin^{6} (2 x)}{\cos^{1} (2 x) \cos^{5} (2 x)}$

Step 5.5.4.1.2

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

$x' = \frac{10 \sin^{4} (2 x)}{\cos^{4} (2 x)} + \frac{10 \sin^{6} (2 x)}{{\cos (2 x)}^{1 + 5}}$

Step 5.5.4.2

Add $1$ and $5$ .

$x' = \frac{10 \sin^{4} (2 x)}{\cos^{4} (2 x)} + \frac{10 \sin^{6} (2 x)}{\cos^{6} (2 x)}$

Step 5.6

Simplify each term.

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Step 5.6.1

Multiply by $1$ .

$x' = \frac{10 (\sin^{4} (2 x) \cdot 1)}{\cos^{4} (2 x)} + \frac{10 \sin^{6} (2 x)}{\cos^{6} (2 x)}$

Step 5.6.2

Multiply by $1$ .

$x' = \frac{10 (\sin^{4} (2 x) \cdot 1)}{\cos^{4} (2 x) \cdot 1} + \frac{10 \sin^{6} (2 x)}{\cos^{6} (2 x)}$

Step 5.6.3

Separate fractions.

$x' = \frac{10 \cdot (1)}{1} \cdot \frac{\sin^{4} (2 x)}{\cos^{4} (2 x)} + \frac{10 \sin^{6} (2 x)}{\cos^{6} (2 x)}$

Step 5.6.4

Convert from $\frac{\sin^{4} (2 x)}{\cos^{4} (2 x)}$ to $\tan^{4} (2 x)$ .

$x' = \frac{10 \cdot (1)}{1} \tan^{4} (2 x) + \frac{10 \sin^{6} (2 x)}{\cos^{6} (2 x)}$

Step 5.6.5

Multiply $10$ by $1$ .

$x' = \frac{10}{1} \tan^{4} (2 x) + \frac{10 \sin^{6} (2 x)}{\cos^{6} (2 x)}$

Step 5.6.6

Divide $10$ by $1$ .

$x' = 10 \tan^{4} (2 x) + \frac{10 \sin^{6} (2 x)}{\cos^{6} (2 x)}$

Step 5.6.7

Multiply by $1$ .

$x' = 10 \tan^{4} (2 x) + \frac{10 (\sin^{6} (2 x) \cdot 1)}{\cos^{6} (2 x)}$

Step 5.6.8

Multiply by $1$ .

$x' = 10 \tan^{4} (2 x) + \frac{10 (\sin^{6} (2 x) \cdot 1)}{\cos^{6} (2 x) \cdot 1}$

Step 5.6.9

Separate fractions.

$x' = 10 \tan^{4} (2 x) + \frac{10 \cdot (1)}{1} \cdot \frac{\sin^{6} (2 x)}{\cos^{6} (2 x)}$

Step 5.6.10

Convert from $\frac{\sin^{6} (2 x)}{\cos^{6} (2 x)}$ to $\tan^{6} (2 x)$ .

$x' = 10 \tan^{4} (2 x) + \frac{10 \cdot (1)}{1} \tan^{6} (2 x)$

Step 5.6.11

Multiply $10$ by $1$ .

$x' = 10 \tan^{4} (2 x) + \frac{10}{1} \tan^{6} (2 x)$

Step 5.6.12

Divide $10$ by $1$ .

$x' = 10 \tan^{4} (2 x) + 10 \tan^{6} (2 x)$