Calculus Examples

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

Step 1

Simplify the numerator.

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Step 1.1

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

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Step 1.1.1

Factor $e^{- t}$ out of $- e^{- t} \cos (2 t)$ .

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

Step 1.1.2

Factor $e^{- t}$ out of $- 2 e^{- t} \sin (2 t)$ .

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

Step 1.1.3

Factor $e^{- t}$ out of $e^{- t} (- 1 \cos (2 t)) + e^{- t} (- 2 \sin (2 t))$ .

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

Step 1.2

Factor $- 1$ out of $- 1 \cos (2 t) - 2 \sin (2 t)$ .

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Step 1.2.1

Factor $- 1$ out of $- 1 \cos (2 t)$ .

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

Step 1.2.2

Factor $- 1$ out of $- 2 \sin (2 t)$ .

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

Step 1.2.3

Factor $- 1$ out of $- 1 (\cos (2 t)) - (2 \sin (2 t))$ .

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

Step 1.2.4

Rewrite $- 1 (\cos (2 t) + 2 \sin (2 t))$ as $- (\cos (2 t) + 2 \sin (2 t))$ .

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

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

Step 1.3

Factor out negative.

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

Step 2

Simplify the denominator.

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Step 2.1

Factor $2 e^{- 2 t}$ out of $- 2 e^{- 2 t} \sin (2 t) + 2 e^{- 2 t} \cos (2 t)$ .

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Step 2.1.1

Factor $2 e^{- 2 t}$ out of $- 2 e^{- 2 t} \sin (2 t)$ .

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

Step 2.1.2

Factor $2 e^{- 2 t}$ out of $2 e^{- 2 t} \cos (2 t)$ .

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

Step 2.1.3

Factor $2 e^{- 2 t}$ out of $2 e^{- 2 t} (- 1 \sin (2 t)) + 2 e^{- 2 t} (\cos (2 t))$ .

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

Step 2.2

Rewrite $- 1 \sin (2 t)$ as $- \sin (2 t)$ .

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

Step 3

Simplify terms.

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Step 3.1

Cancel the common factor of $e^{- t}$ and $e^{- 2 t}$ .

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Step 3.1.1

Factor $e^{- 2 t}$ out of $- e^{- t} (\cos (2 t) + 2 \sin (2 t))$ .

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

Step 3.1.2

Cancel the common factors.

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Step 3.1.2.1

Factor $e^{- 2 t}$ out of $2 e^{- 2 t} (- \sin (2 t) + \cos (2 t))$ .

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

Step 3.1.2.2

Cancel the common factor.

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

Step 3.1.2.3

Rewrite the expression.

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

Step 3.2

Move the negative in front of the fraction.

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

Step 3.3

Factor $- 1$ out of $- \sin (2 t)$ .

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

Step 3.4

Factor $- 1$ out of $\cos (2 t)$ .

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

Step 3.5

Factor $- 1$ out of $- (\sin (2 t)) - 1 (- \cos (2 t))$ .

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

Step 3.6

Simplify the expression.

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Step 3.6.1

Rewrite $- (\sin (2 t) - \cos (2 t))$ as $- 1 (\sin (2 t) - \cos (2 t))$ .

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

Step 3.6.2

Move the negative in front of the fraction.

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

Step 3.6.3

Multiply $- 1$ by $- 1$ .

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

Step 3.6.4

Multiply $\frac{e^{t} (\cos (2 t) + 2 \sin (2 t))}{2 (\sin (2 t) - \cos (2 t))}$ by $1$ .

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