Calculus Examples

$y = (t^{3} - 7 t^{2} + 1) e^{t}$

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 $f (t) = t^{3} - 7 t^{2} + 1$ and $g (t) = e^{t}$ .

$(t^{3} - 7 t^{2} + 1) \frac{d}{d t} [e^{t}] + e^{t} \frac{d}{d t} [t^{3} - 7 t^{2} + 1]$

Step 2

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

$(t^{3} - 7 t^{2} + 1) e^{t} + e^{t} \frac{d}{d t} [t^{3} - 7 t^{2} + 1]$

Step 3

Differentiate.

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Step 3.1

By the Sum Rule, the derivative of $t^{3} - 7 t^{2} + 1$ with respect to $t$ is $\frac{d}{d t} [t^{3}] + \frac{d}{d t} [- 7 t^{2}] + \frac{d}{d t} [1]$ .

$(t^{3} - 7 t^{2} + 1) e^{t} + e^{t} (\frac{d}{d t} [t^{3}] + \frac{d}{d t} [- 7 t^{2}] + \frac{d}{d t} [1])$

Step 3.2

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 3$ .

$(t^{3} - 7 t^{2} + 1) e^{t} + e^{t} (3 t^{2} + \frac{d}{d t} [- 7 t^{2}] + \frac{d}{d t} [1])$

Step 3.3

Since $- 7$ is constant with respect to $t$ , the derivative of $- 7 t^{2}$ with respect to $t$ is $- 7 \frac{d}{d t} [t^{2}]$ .

$(t^{3} - 7 t^{2} + 1) e^{t} + e^{t} (3 t^{2} - 7 \frac{d}{d t} [t^{2}] + \frac{d}{d t} [1])$

Step 3.4

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 2$ .

$(t^{3} - 7 t^{2} + 1) e^{t} + e^{t} (3 t^{2} - 7 (2 t) + \frac{d}{d t} [1])$

Step 3.5

Multiply $2$ by $- 7$ .

$(t^{3} - 7 t^{2} + 1) e^{t} + e^{t} (3 t^{2} - 14 t + \frac{d}{d t} [1])$

Step 3.6

Since $1$ is constant with respect to $t$ , the derivative of $1$ with respect to $t$ is $0$ .

$(t^{3} - 7 t^{2} + 1) e^{t} + e^{t} (3 t^{2} - 14 t + 0)$

Step 3.7

Add $3 t^{2} - 14 t$ and $0$ .

$(t^{3} - 7 t^{2} + 1) e^{t} + e^{t} (3 t^{2} - 14 t)$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$t^{3} e^{t} - 7 t^{2} e^{t} + 1 e^{t} + e^{t} (3 t^{2} - 14 t)$

Step 4.2

Apply the distributive property.

$t^{3} e^{t} - 7 t^{2} e^{t} + 1 e^{t} + e^{t} (3 t^{2}) + e^{t} (- 14 t)$

Step 4.3

Combine terms.

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Step 4.3.1

Multiply $e^{t}$ by $1$ .

$t^{3} e^{t} - 7 t^{2} e^{t} + e^{t} + e^{t} (3 t^{2}) + e^{t} (- 14 t)$

Step 4.3.2

Add $- 7 t^{2} e^{t}$ and $e^{t} (3 t^{2})$ .

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Step 4.3.2.1

Move $e^{t}$ .

$t^{3} e^{t} - 7 t^{2} e^{t} + 3 t^{2} e^{t} + e^{t} + e^{t} (- 14 t)$

Step 4.3.2.2

Add $- 7 t^{2} e^{t}$ and $3 t^{2} e^{t}$ .

$t^{3} e^{t} - 4 t^{2} e^{t} + e^{t} + e^{t} (- 14 t)$

Step 4.4

Reorder terms.

$e^{t} t^{3} - 4 e^{t} t^{2} - 14 e^{t} t + e^{t}$

Step 4.5

Reorder factors in $e^{t} t^{3} - 4 e^{t} t^{2} - 14 e^{t} t + e^{t}$ .

$t^{3} e^{t} - 4 t^{2} e^{t} - 14 t e^{t} + e^{t}$