Calculus Examples

$y = e^{t} (\cos (2 t) + 2 \sin (2 t))$

Step 1

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 $f (t) = e^{t}$ and $g (t) = \cos (2 t) + 2 \sin (2 t)$ .

$e^{t} \frac{d}{d t} [\cos (2 t) + 2 \sin (2 t)] + (\cos (2 t) + 2 \sin (2 t)) \frac{d}{d t} [e^{t}]$

Step 2

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

$e^{t} (\frac{d}{d t} [\cos (2 t)] + \frac{d}{d t} [2 \sin (2 t)]) + (\cos (2 t) + 2 \sin (2 t)) \frac{d}{d t} [e^{t}]$

Step 3

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

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Step 3.1

To apply the Chain Rule, set $u_{1}$ as $2 t$ .

$e^{t} (\frac{d}{d u_{1}} [\cos (u_{1})] \frac{d}{d t} [2 t] + \frac{d}{d t} [2 \sin (2 t)]) + (\cos (2 t) + 2 \sin (2 t)) \frac{d}{d t} [e^{t}]$

Step 3.2

The derivative of $\cos (u_{1})$ with respect to $u_{1}$ is $- \sin (u_{1})$ .

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

Step 3.3

Replace all occurrences of $u_{1}$ with $2 t$ .

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

Step 4

Differentiate.

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Step 4.1

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

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

Step 4.2

Multiply $2$ by $- 1$ .

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

Step 4.3

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

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

Step 4.4

Multiply $- 2$ by $1$ .

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

Step 4.5

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

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

Step 5

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

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Step 5.1

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

$e^{t} (- 2 \sin (2 t) + 2 (\frac{d}{d u_{2}} [\sin (u_{2})] \frac{d}{d t} [2 t])) + (\cos (2 t) + 2 \sin (2 t)) \frac{d}{d t} [e^{t}]$

Step 5.2

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

$e^{t} (- 2 \sin (2 t) + 2 (\cos (u_{2}) \frac{d}{d t} [2 t])) + (\cos (2 t) + 2 \sin (2 t)) \frac{d}{d t} [e^{t}]$

Step 5.3

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

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

Step 6

Differentiate.

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Step 6.1

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

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

Step 6.2

Multiply $2$ by $2$ .

$e^{t} (- 2 \sin (2 t) + 4 \cos (2 t) \frac{d}{d t} [t]) + (\cos (2 t) + 2 \sin (2 t)) \frac{d}{d t} [e^{t}]$

Step 6.3

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

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

Step 6.4

Multiply $4$ by $1$ .

$e^{t} (- 2 \sin (2 t) + 4 \cos (2 t)) + (\cos (2 t) + 2 \sin (2 t)) \frac{d}{d t} [e^{t}]$

Step 7

Differentiate using the Exponential Rule which states that $\frac{d}{d t} [a^{t}]$ is $a^{t} \ln (a)$ where $a$ = $e$ .

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

Step 8

Simplify.

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Step 8.1

Apply the distributive property.

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

Step 8.2

Apply the distributive property.

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

Step 8.3

Combine terms.

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Step 8.3.1

Reorder $\cos (2 t)$ and $e^{t}$ .

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

Step 8.3.2

Add $4 e^{t} \cos (2 t)$ and $e^{t} \cos (2 t)$ .

$e^{t} (- 2 \sin (2 t)) + 5 e^{t} \cos (2 t) + 2 \sin (2 t) e^{t}$

Step 8.3.3

Move $\sin (2 t)$ .

$5 e^{t} \cos (2 t) - 2 \cdot e^{t} \sin (2 t) + 2 e^{t} \sin (2 t)$

Step 8.3.4

Add $- 2 e^{t} \sin (2 t)$ and $2 e^{t} \sin (2 t)$ .

$5 e^{t} \cos (2 t) + 0$

Step 8.3.5

Add $5 e^{t} \cos (2 t)$ and $0$ .

$5 e^{t} \cos (2 t)$