Calculus Examples

$\frac{e^{- t}}{1 + t^{2}}$

Step 1

Differentiate using the Quotient Rule which states that $\frac{d}{d t} [\frac{f (t)}{g (t)}]$ is $\frac{g (t) \frac{d}{d t} [f (t)] - f (t) \frac{d}{d t} [g (t)]}{{g (t)}^{2}}$ where $f (t) = e^{- t}$ and $g (t) = 1 + t^{2}$ .

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

Step 2

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 $g (t) = - t$ .

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 3

Differentiate.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Simplify the expression.

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Step 3.3.1

Multiply $- 1$ by $1$ .

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

Step 3.3.2

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

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

Step 3.3.3

Rewrite $- 1 (1 + t^{2})$ as $- (1 + t^{2})$ .

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

Step 3.4

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

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

Step 3.5

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

$\frac{- (1 + t^{2}) e^{- t} - e^{- t} (0 + \frac{d}{d t} [t^{2}])}{{(1 + t^{2})}^{2}}$

Step 3.6

Add $0$ and $\frac{d}{d t} [t^{2}]$ .

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

Step 3.7

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

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

Step 3.8

Multiply $2$ by $- 1$ .

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

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Apply the distributive property.

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

Step 4.3

Simplify the numerator.

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Step 4.3.1

Simplify each term.

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Step 4.3.1.1

Multiply $- 1$ by $1$ .

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

Step 4.3.1.2

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

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

Step 4.3.2

Reorder factors in $- e^{- t} - t^{2} e^{- t} - 2 e^{- t} t$ .

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

Step 4.4

Reorder terms.

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

Step 4.5

Simplify the numerator.

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Step 4.5.1

Factor $e^{- t}$ out of $- e^{- t} t^{2} - 2 e^{- t} t - e^{- t}$ .

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Step 4.5.1.1

Factor $e^{- t}$ out of $- e^{- t} t^{2}$ .

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

Step 4.5.1.2

Factor $e^{- t}$ out of $- 2 e^{- t} t$ .

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

Step 4.5.1.3

Factor $e^{- t}$ out of $- e^{- t}$ .

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

Step 4.5.1.4

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

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

Step 4.5.1.5

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

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

Step 4.5.2

Factor by grouping.

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Step 4.5.2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot - 1 = 1$ and whose sum is $b = - 2$ .

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Step 4.5.2.1.1

Factor $- 2$ out of $- 2 t$ .

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

Step 4.5.2.1.2

Rewrite $- 2$ as $- 1$ plus $- 1$

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

Step 4.5.2.1.3

Apply the distributive property.

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

Step 4.5.2.2

Factor out the greatest common factor from each group.

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Step 4.5.2.2.1

Group the first two terms and the last two terms.

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

Step 4.5.2.2.2

Factor out the greatest common factor (GCF) from each group.

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

Step 4.5.2.3

Factor the polynomial by factoring out the greatest common factor, $- t - 1$ .

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

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

Step 4.5.3

Combine exponents.

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Step 4.5.3.1

Factor $- 1$ out of $- t$ .

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

Step 4.5.3.2

Rewrite $- 1$ as $- 1 (1)$ .

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

Step 4.5.3.3

Factor $- 1$ out of $- (t) - 1 (1)$ .

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

Step 4.5.3.4

Rewrite $- (t + 1)$ as $- 1 (t + 1)$ .

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

Step 4.5.3.5

Raise $t + 1$ to the power of $1$ .

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

Step 4.5.3.6

Raise $t + 1$ to the power of $1$ .

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

Step 4.5.3.7

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

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

Step 4.5.3.8

Add $1$ and $1$ .

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

Step 4.5.4

Factor out negative.

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

Step 4.6

Move the negative in front of the fraction.

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