Calculus Examples

$\frac{\sin (t)}{2 \sin (2 t)}$

Step 1

Since $\frac{1}{2}$ is constant with respect to $t$ , the derivative of $\frac{\sin (t)}{2 \sin (2 t)}$ with respect to $t$ is $\frac{1}{2} \frac{d}{d t} [\frac{\sin (t)}{\sin (2 t)}]$ .

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

Step 2

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

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

Step 3

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

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

Step 4

Simplify.

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Step 4.1

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

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

Step 4.2

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

$\frac{1}{2} \frac{d}{d t} [\sin (t) \csc (2 t)]$

Step 5

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

$\frac{1}{2} (\sin (t) \frac{d}{d t} [\csc (2 t)] + \csc (2 t) \frac{d}{d t} [\sin (t)])$

Step 6

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = \csc (t)$ and $g (t) = 2 t$ .

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Step 6.1

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

$\frac{1}{2} (\sin (t) (\frac{d}{d u} [\csc (u)] \frac{d}{d t} [2 t]) + \csc (2 t) \frac{d}{d t} [\sin (t)])$

Step 6.2

The derivative of $\csc (u)$ with respect to $u$ is $- \csc (u) \cot (u)$ .

$\frac{1}{2} (\sin (t) (- \csc (u) \cot (u) \frac{d}{d t} [2 t]) + \csc (2 t) \frac{d}{d t} [\sin (t)])$

Step 6.3

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

$\frac{1}{2} (\sin (t) (- \csc (2 t) \cot (2 t) \frac{d}{d t} [2 t]) + \csc (2 t) \frac{d}{d t} [\sin (t)])$

Step 7

Differentiate.

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Step 7.1

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

$\frac{1}{2} (\sin (t) (- \csc (2 t) \cot (2 t) (2 \frac{d}{d t} [t])) + \csc (2 t) \frac{d}{d t} [\sin (t)])$

Step 7.2

Multiply $2$ by $- 1$ .

$\frac{1}{2} (\sin (t) (- 2 \csc (2 t) \cot (2 t) \frac{d}{d t} [t]) + \csc (2 t) \frac{d}{d t} [\sin (t)])$

Step 7.3

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

$\frac{1}{2} (\sin (t) (- 2 \csc (2 t) \cot (2 t) \cdot 1) + \csc (2 t) \frac{d}{d t} [\sin (t)])$

Step 7.4

Multiply $- 2$ by $1$ .

$\frac{1}{2} (\sin (t) (- 2 \csc (2 t) \cot (2 t)) + \csc (2 t) \frac{d}{d t} [\sin (t)])$

Step 8

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

$\frac{1}{2} (\sin (t) (- 2 \csc (2 t) \cot (2 t)) + \csc (2 t) \cos (t))$

Step 9

Simplify.

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Step 9.1

Apply the distributive property.

$\frac{1}{2} (\sin (t) (- 2 \csc (2 t) \cot (2 t))) + \frac{1}{2} (\csc (2 t) \cos (t))$

Step 9.2

Combine terms.

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Step 9.2.1

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

$\frac{\sin (t)}{2} (- 2 \csc (2 t) \cot (2 t)) + \frac{1}{2} (\csc (2 t) \cos (t))$

Step 9.2.2

Combine $- 2$ and $\frac{\sin (t)}{2}$ .

$\frac{- 2 \sin (t)}{2} (\csc (2 t) \cot (2 t)) + \frac{1}{2} (\csc (2 t) \cos (t))$

Step 9.2.3

Combine $\csc (2 t)$ and $\frac{- 2 \sin (t)}{2}$ .

$\frac{\csc (2 t) (- 2 \sin (t))}{2} \cot (2 t) + \frac{1}{2} (\csc (2 t) \cos (t))$

Step 9.2.4

Combine $\frac{\csc (2 t) (- 2 \sin (t))}{2}$ and $\cot (2 t)$ .

$\frac{\csc (2 t) (- 2 \sin (t)) \cot (2 t)}{2} + \frac{1}{2} (\csc (2 t) \cos (t))$

Step 9.2.5

Cancel the common factor of $- 2$ and $2$ .

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Step 9.2.5.1

Factor $2$ out of $\csc (2 t) (- 2 \sin (t)) \cot (2 t)$ .

$\frac{2 (\csc (2 t) (- \sin (t)) \cot (2 t))}{2} + \frac{1}{2} (\csc (2 t) \cos (t))$

Step 9.2.5.2

Cancel the common factors.

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Step 9.2.5.2.1

Factor $2$ out of $2$ .

$\frac{2 (\csc (2 t) (- \sin (t)) \cot (2 t))}{2 (1)} + \frac{1}{2} (\csc (2 t) \cos (t))$

Step 9.2.5.2.2

Cancel the common factor.

$\frac{2 (\csc (2 t) (- \sin (t)) \cot (2 t))}{2 \cdot 1} + \frac{1}{2} (\csc (2 t) \cos (t))$

Step 9.2.5.2.3

Rewrite the expression.

$\frac{\csc (2 t) (- \sin (t)) \cot (2 t)}{1} + \frac{1}{2} (\csc (2 t) \cos (t))$

Step 9.2.5.2.4

Divide $\csc (2 t) (- \sin (t)) \cot (2 t)$ by $1$ .

$\csc (2 t) (- \sin (t)) \cot (2 t) + \frac{1}{2} (\csc (2 t) \cos (t))$

Step 9.2.6

Combine $\csc (2 t)$ and $\frac{1}{2}$ .

$\csc (2 t) (- \sin (t)) \cot (2 t) + \frac{\csc (2 t)}{2} \cos (t)$

Step 9.2.7

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

$\csc (2 t) (- \sin (t)) \cot (2 t) + \frac{\csc (2 t) \cos (t)}{2}$

Step 9.3

Reorder terms.

$- \cot (2 t) \csc (2 t) \sin (t) + \frac{\csc (2 t) \cos (t)}{2}$

Step 9.4

Simplify each term.

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Step 9.4.1

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

$- \frac{\cos (2 t)}{\sin (2 t)} \csc (2 t) \sin (t) + \frac{\csc (2 t) \cos (t)}{2}$

Step 9.4.2

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

$- \frac{\cos (2 t)}{\sin (2 t)} \cdot \frac{1}{\sin (2 t)} \sin (t) + \frac{\csc (2 t) \cos (t)}{2}$

Step 9.4.3

Multiply $- \frac{\cos (2 t)}{\sin (2 t)} \cdot \frac{1}{\sin (2 t)}$ .

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Step 9.4.3.1

Multiply $\frac{1}{\sin (2 t)}$ by $\frac{\cos (2 t)}{\sin (2 t)}$ .

$- \frac{\cos (2 t)}{\sin (2 t) \sin (2 t)} \sin (t) + \frac{\csc (2 t) \cos (t)}{2}$

Step 9.4.3.2

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

$- \frac{\cos (2 t)}{\sin^{1} (2 t) \sin (2 t)} \sin (t) + \frac{\csc (2 t) \cos (t)}{2}$

Step 9.4.3.3

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

$- \frac{\cos (2 t)}{\sin^{1} (2 t) \sin^{1} (2 t)} \sin (t) + \frac{\csc (2 t) \cos (t)}{2}$

Step 9.4.3.4

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

$- \frac{\cos (2 t)}{{\sin (2 t)}^{1 + 1}} \sin (t) + \frac{\csc (2 t) \cos (t)}{2}$

Step 9.4.3.5

Add $1$ and $1$ .

$- \frac{\cos (2 t)}{\sin^{2} (2 t)} \sin (t) + \frac{\csc (2 t) \cos (t)}{2}$

Step 9.4.4

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

$- \frac{\sin (t) \cos (2 t)}{\sin^{2} (2 t)} + \frac{\csc (2 t) \cos (t)}{2}$

Step 9.4.5

Use the double-angle identity to transform $\cos (2 x)$ to $2 \cos^{2} (x) - 1$ .

$- \frac{\sin (t) (2 \cos^{2} (t) - 1)}{\sin^{2} (2 t)} + \frac{\csc (2 t) \cos (t)}{2}$

Step 9.4.6

Simplify the denominator.

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Step 9.4.6.1

Apply the sine double-angle identity.

$- \frac{\sin (t) (2 \cos^{2} (t) - 1)}{{(2 \sin (t) \cos (t))}^{2}} + \frac{\csc (2 t) \cos (t)}{2}$

Step 9.4.6.2

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 9.4.6.2.1

Apply the product rule to $2 \sin (t) \cos (t)$ .

$- \frac{\sin (t) (2 \cos^{2} (t) - 1)}{{(2 \sin (t))}^{2} \cos^{2} (t)} + \frac{\csc (2 t) \cos (t)}{2}$

Step 9.4.6.2.2

Apply the product rule to $2 \sin (t)$ .

$- \frac{\sin (t) (2 \cos^{2} (t) - 1)}{2^{2} \sin^{2} (t) \cos^{2} (t)} + \frac{\csc (2 t) \cos (t)}{2}$

Step 9.4.6.3

Raise $2$ to the power of $2$ .

$- \frac{\sin (t) (2 \cos^{2} (t) - 1)}{4 \sin^{2} (t) \cos^{2} (t)} + \frac{\csc (2 t) \cos (t)}{2}$

Step 9.4.7

Cancel the common factors.

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Step 9.4.7.1

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

$- \frac{\sin (t) (2 \cos^{2} (t) - 1)}{\sin (t) (4 \sin (t) \cos^{2} (t))} + \frac{\csc (2 t) \cos (t)}{2}$

Step 9.4.7.2

Cancel the common factor.

$- \frac{\sin (t) (2 \cos^{2} (t) - 1)}{\sin (t) (4 \sin (t) \cos^{2} (t))} + \frac{\csc (2 t) \cos (t)}{2}$

Step 9.4.7.3

Rewrite the expression.

$- \frac{2 \cos^{2} (t) - 1}{4 \sin (t) \cos^{2} (t)} + \frac{\csc (2 t) \cos (t)}{2}$

Step 9.4.8

Apply the cosine double-angle identity.

$- \frac{\cos (2 t)}{4 \sin (t) \cos^{2} (t)} + \frac{\csc (2 t) \cos (t)}{2}$

Step 9.4.9

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

$- \frac{\cos (2 t)}{4 \sin (t) \cos^{2} (t)} + \frac{\frac{1}{\sin (2 t)} \cos (t)}{2}$

Step 9.4.10

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

$- \frac{\cos (2 t)}{4 \sin (t) \cos^{2} (t)} + \frac{\frac{\cos (t)}{\sin (2 t)}}{2}$

Step 9.4.11

Simplify the numerator.

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Step 9.4.11.1

Apply the sine double-angle identity.

$- \frac{\cos (2 t)}{4 \sin (t) \cos^{2} (t)} + \frac{\frac{\cos (t)}{2 \sin (t) \cos (t)}}{2}$

Step 9.4.11.2

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

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Step 9.4.11.2.1

Cancel the common factor.

$- \frac{\cos (2 t)}{4 \sin (t) \cos^{2} (t)} + \frac{\frac{\cos (t)}{2 \sin (t) \cos (t)}}{2}$

Step 9.4.11.2.2

Rewrite the expression.

$- \frac{\cos (2 t)}{4 \sin (t) \cos^{2} (t)} + \frac{\frac{1}{2 \sin (t)}}{2}$

Step 9.4.12

Multiply the numerator by the reciprocal of the denominator.

$- \frac{\cos (2 t)}{4 \sin (t) \cos^{2} (t)} + \frac{1}{2 \sin (t)} \cdot \frac{1}{2}$

Step 9.4.13

Multiply $\frac{1}{2 \sin (t)} \cdot \frac{1}{2}$ .

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Step 9.4.13.1

Multiply $\frac{1}{2 \sin (t)}$ by $\frac{1}{2}$ .

$- \frac{\cos (2 t)}{4 \sin (t) \cos^{2} (t)} + \frac{1}{2 \sin (t) \cdot 2}$

Step 9.4.13.2

Multiply $2$ by $2$ .

$- \frac{\cos (2 t)}{4 \sin (t) \cos^{2} (t)} + \frac{1}{4 \sin (t)}$

Step 9.5

Simplify each term.

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Step 9.5.1

Separate fractions.

$- (\frac{1}{4 \cos^{2} (t)} \cdot \frac{\cos (2 t)}{\sin (t)}) + \frac{1}{4 \sin (t)}$

Step 9.5.2

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

$- (\frac{1}{4 \cos^{2} (t)} (\cos (2 t) \frac{1}{\sin (t)})) + \frac{1}{4 \sin (t)}$

Step 9.5.3

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

$- (\frac{1}{4 \cos^{2} (t)} (\frac{\cos (2 t)}{1} \cdot \frac{1}{\sin (t)})) + \frac{1}{4 \sin (t)}$

Step 9.5.4

Simplify.

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Step 9.5.4.1

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

$- (\frac{1}{4 \cos^{2} (t)} (\cos (2 t) \frac{1}{\sin (t)})) + \frac{1}{4 \sin (t)}$

Step 9.5.4.2

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

$- (\frac{1}{4 \cos^{2} (t)} (\cos (2 t) \csc (t))) + \frac{1}{4 \sin (t)}$

Step 9.5.5

Multiply by $1$ .

$- (\frac{1}{4 (\cos^{2} (t) \cdot 1)} (\cos (2 t) \csc (t))) + \frac{1}{4 \sin (t)}$

Step 9.5.6

Separate fractions.

$- (\frac{1}{4 \cdot (1)} \cdot \frac{1}{\cos^{2} (t)} (\cos (2 t) \csc (t))) + \frac{1}{4 \sin (t)}$

Step 9.5.7

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

$- (\frac{1}{4 \cdot (1)} \sec^{2} (t) (\cos (2 t) \csc (t))) + \frac{1}{4 \sin (t)}$

Step 9.5.8

Multiply $4$ by $1$ .

$- (\frac{1}{4} \sec^{2} (t) (\cos (2 t) \csc (t))) + \frac{1}{4 \sin (t)}$

Step 9.5.9

Combine $\frac{1}{4}$ and $\sec^{2} (t)$ .

$- (\frac{\sec^{2} (t)}{4} (\cos (2 t) \csc (t))) + \frac{1}{4 \sin (t)}$

Step 9.5.10

Multiply $\frac{\sec^{2} (t)}{4} (\cos (2 t) \csc (t))$ .

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Step 9.5.10.1

Combine $\cos (2 t)$ and $\frac{\sec^{2} (t)}{4}$ .

$- (\frac{\cos (2 t) \sec^{2} (t)}{4} \csc (t)) + \frac{1}{4 \sin (t)}$

Step 9.5.10.2

Combine $\frac{\cos (2 t) \sec^{2} (t)}{4}$ and $\csc (t)$ .

$- \frac{\cos (2 t) \sec^{2} (t) \csc (t)}{4} + \frac{1}{4 \sin (t)}$

Step 9.5.11

Separate fractions.

$- \frac{\cos (2 t) \sec^{2} (t) \csc (t)}{4} + \frac{1}{4} \cdot \frac{1}{\sin (t)}$

Step 9.5.12

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

$- \frac{\cos (2 t) \sec^{2} (t) \csc (t)}{4} + \frac{1}{4} \csc (t)$

Step 9.5.13

Combine $\frac{1}{4}$ and $\csc (t)$ .

$- \frac{\cos (2 t) \sec^{2} (t) \csc (t)}{4} + \frac{\csc (t)}{4}$