Calculus Examples

$y = 4 e^{t} (e^{4 t} - e^{t})$

Step 1

Since $4$ is constant with respect to $t$ , the derivative of $4 e^{t} (e^{4 t} - e^{t})$ with respect to $t$ is $4 \frac{d}{d t} [e^{t} (e^{4 t} - e^{t})]$ .

$4 \frac{d}{d t} [e^{t} (e^{4 t} - e^{t})]$

Step 2

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 $g (t) = e^{4 t} - e^{t}$ .

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

Step 3

By the Sum Rule, the derivative of $e^{4 t} - e^{t}$ with respect to $t$ is $\frac{d}{d t} [e^{4 t}] + \frac{d}{d t} [- e^{t}]$ .

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

Step 4

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = e^{t}$ and $g (t) = 4 t$ .

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Step 4.1

To apply the Chain Rule, set $u$ as $4 t$ .

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

Step 4.2

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

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

Step 4.3

Replace all occurrences of $u$ with $4 t$ .

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

Step 5

Differentiate.

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Step 5.1

Since $4$ is constant with respect to $t$ , the derivative of $4 t$ with respect to $t$ is $4 \frac{d}{d t} [t]$ .

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

Step 5.2

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

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

Step 5.3

Simplify the expression.

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Step 5.3.1

Multiply $4$ by $1$ .

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

Step 5.3.2

Move $4$ to the left of $e^{4 t}$ .

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

Step 5.4

Since $- 1$ is constant with respect to $t$ , the derivative of $- e^{t}$ with respect to $t$ is $- \frac{d}{d t} [e^{t}]$ .

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

Step 6

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

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

Step 7

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

$4 (e^{t} (4 e^{4 t} - e^{t}) + (e^{4 t} - e^{t}) e^{t})$

Step 8

Simplify.

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Step 8.1

Apply the distributive property.

$4 (e^{t} (4 e^{4 t}) + e^{t} (- e^{t}) + (e^{4 t} - e^{t}) e^{t})$

Step 8.2

Apply the distributive property.

$4 (e^{t} (4 e^{4 t}) + e^{t} (- e^{t}) + e^{4 t} e^{t} - e^{t} e^{t})$

Step 8.3

Apply the distributive property.

$4 (e^{t} (4 e^{4 t})) + 4 (e^{t} (- e^{t})) + 4 (e^{4 t} e^{t}) + 4 (- e^{t} e^{t})$

Step 8.4

Combine terms.

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Step 8.4.1

Multiply $e^{t}$ by $e^{4 t}$ by adding the exponents.

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Step 8.4.1.1

Move $e^{4 t}$ .

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

Step 8.4.1.2

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

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

Step 8.4.1.3

Add $4 t$ and $t$ .

$4 (e^{5 t} \cdot 4) + 4 (e^{t} (- e^{t})) + 4 (e^{4 t} e^{t}) + 4 (- e^{t} e^{t})$

Step 8.4.2

Move $4$ to the left of $e^{5 t}$ .

$4 (4 \cdot e^{5 t}) + 4 (e^{t} (- e^{t})) + 4 (e^{4 t} e^{t}) + 4 (- e^{t} e^{t})$

Step 8.4.3

Multiply $4$ by $4$ .

$16 e^{5 t} + 4 (e^{t} (- e^{t})) + 4 (e^{4 t} e^{t}) + 4 (- e^{t} e^{t})$

Step 8.4.4

Multiply $e^{t}$ by $e^{t}$ by adding the exponents.

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Step 8.4.4.1

Move $e^{t}$ .

$16 e^{5 t} + 4 (e^{t} e^{t} \cdot - 1) + 4 (e^{4 t} e^{t}) + 4 (- e^{t} e^{t})$

Step 8.4.4.2

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

$16 e^{5 t} + 4 (e^{t + t} \cdot - 1) + 4 (e^{4 t} e^{t}) + 4 (- e^{t} e^{t})$

Step 8.4.4.3

Add $t$ and $t$ .

$16 e^{5 t} + 4 (e^{2 t} \cdot - 1) + 4 (e^{4 t} e^{t}) + 4 (- e^{t} e^{t})$

Step 8.4.5

Move $- 1$ to the left of $e^{2 t}$ .

$16 e^{5 t} + 4 (- 1 \cdot e^{2 t}) + 4 (e^{4 t} e^{t}) + 4 (- e^{t} e^{t})$

Step 8.4.6

Rewrite $- 1 e^{2 t}$ as $- e^{2 t}$ .

$16 e^{5 t} + 4 (- e^{2 t}) + 4 (e^{4 t} e^{t}) + 4 (- e^{t} e^{t})$

Step 8.4.7

Multiply $- 1$ by $4$ .

$16 e^{5 t} - 4 e^{2 t} + 4 (e^{4 t} e^{t}) + 4 (- e^{t} e^{t})$

Step 8.4.8

Multiply $e^{4 t}$ by $e^{t}$ by adding the exponents.

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Step 8.4.8.1

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

$16 e^{5 t} - 4 e^{2 t} + 4 e^{4 t + t} + 4 (- e^{t} e^{t})$

Step 8.4.8.2

Add $4 t$ and $t$ .

$16 e^{5 t} - 4 e^{2 t} + 4 e^{5 t} + 4 (- e^{t} e^{t})$

Step 8.4.9

Multiply $e^{t}$ by $e^{t}$ by adding the exponents.

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Step 8.4.9.1

Move $e^{t}$ .

$16 e^{5 t} - 4 e^{2 t} + 4 e^{5 t} + 4 (- (e^{t} e^{t}))$

Step 8.4.9.2

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

$16 e^{5 t} - 4 e^{2 t} + 4 e^{5 t} + 4 (- e^{t + t})$

Step 8.4.9.3

Add $t$ and $t$ .

$16 e^{5 t} - 4 e^{2 t} + 4 e^{5 t} + 4 (- e^{2 t})$

Step 8.4.10

Multiply $- 1$ by $4$ .

$16 e^{5 t} - 4 e^{2 t} + 4 e^{5 t} - 4 e^{2 t}$

Step 8.4.11

Add $16 e^{5 t}$ and $4 e^{5 t}$ .

$20 e^{5 t} - 4 e^{2 t} - 4 e^{2 t}$

Step 8.4.12

Subtract $4 e^{2 t}$ from $- 4 e^{2 t}$ .

$20 e^{5 t} - 8 e^{2 t}$