Calculus Examples

$\frac{e^{x}}{x^{2} + 1}$

Step 1

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

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

Step 2

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

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

Step 3

Differentiate.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

$\frac{(x^{2} + 1) e^{x} - e^{x} (2 x + 0)}{{(x^{2} + 1)}^{2}}$

Step 3.4

Simplify the expression.

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Step 3.4.1

Add $2 x$ and $0$ .

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

Step 3.4.2

Multiply $2$ by $- 1$ .

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

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Simplify the numerator.

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Step 4.2.1

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

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

Step 4.2.2

Reorder factors in $x^{2} e^{x} + e^{x} - 2 e^{x} x$ .

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

Step 4.3

Reorder terms.

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

Step 4.4

Simplify the numerator.

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Step 4.4.1

Factor by grouping.

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Step 4.4.1.1

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 4.4.1.1.1

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

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

Step 4.4.1.1.2

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

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

Step 4.4.1.1.3

Apply the distributive property.

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

Step 4.4.1.1.4

Move parentheses.

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

Step 4.4.1.2

Factor out the greatest common factor from each group.

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Step 4.4.1.2.1

Group the first two terms and the last two terms.

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

Step 4.4.1.2.2

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

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

Step 4.4.1.3

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

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

Step 4.4.2

Factor $e^{x}$ out of $e^{x} x - 1 e^{x}$ .

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Step 4.4.2.1

Factor $e^{x}$ out of $e^{x} x$ .

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

Step 4.4.2.2

Factor $e^{x}$ out of $- 1 e^{x}$ .

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

Step 4.4.2.3

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

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

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

Step 4.4.3

Combine exponents.

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Step 4.4.3.1

Raise $x - 1$ to the power of $1$ .

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

Step 4.4.3.2

Raise $x - 1$ to the power of $1$ .

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

Step 4.4.3.3

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

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

Step 4.4.3.4

Add $1$ and $1$ .

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

Step 4.5

Reorder factors in $\frac{{(x - 1)}^{2} e^{x}}{{(x^{2} + 1)}^{2}}$ .

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