Calculus Examples

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

Step 1

Since $2$ is constant with respect to $x$ , the derivative of $\frac{2 e^{x}}{x^{2} + 1}$ with respect to $x$ is $2 \frac{d}{d x} [\frac{e^{x}}{x^{2} + 1}]$ .

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

Step 2

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 $g (x) = x^{2} + 1$ .

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

Step 3

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

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

Step 4

Differentiate.

Tap for more steps...

Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

Combine fractions.

Tap for more steps...

Step 4.4.1

Add $2 x$ and $0$ .

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

Step 4.4.2

Multiply $2$ by $- 1$ .

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

Step 4.4.3

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

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

Step 5

Simplify.

Tap for more steps...

Step 5.1

Apply the distributive property.

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

Step 5.2

Apply the distributive property.

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

Step 5.3

Simplify the numerator.

Tap for more steps...

Step 5.3.1

Simplify each term.

Tap for more steps...

Step 5.3.1.1

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

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

Step 5.3.1.2

Multiply $- 2$ by $2$ .

$\frac{2 x^{2} e^{x} + 2 e^{x} - 4 e^{x} x}{{(x^{2} + 1)}^{2}}$

Step 5.3.2

Reorder factors in $2 x^{2} e^{x} + 2 e^{x} - 4 e^{x} x$ .

$\frac{2 x^{2} e^{x} + 2 e^{x} - 4 x e^{x}}{{(x^{2} + 1)}^{2}}$

Step 5.4

Reorder terms.

$\frac{2 e^{x} x^{2} - 4 e^{x} x + 2 e^{x}}{{(x^{2} + 1)}^{2}}$

Step 5.5

Simplify the numerator.

Tap for more steps...

Step 5.5.1

Factor $2 e^{x}$ out of $2 e^{x} x^{2} - 4 e^{x} x + 2 e^{x}$ .

Tap for more steps...

Step 5.5.1.1

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

$\frac{2 e^{x} (x^{2}) - 4 e^{x} x + 2 e^{x}}{{(x^{2} + 1)}^{2}}$

Step 5.5.1.2

Factor $2 e^{x}$ out of $- 4 e^{x} x$ .

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

Step 5.5.1.3

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

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

Step 5.5.1.4

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

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

Step 5.5.1.5

Factor $2 e^{x}$ out of $2 e^{x} (x^{2} - 2 x) + 2 e^{x} (1)$ .

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

Step 5.5.2

Factor using the perfect square rule.

Tap for more steps...

Step 5.5.2.1

Rewrite $1$ as $1^{2}$ .

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

Step 5.5.2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 x = 2 \cdot x \cdot 1$

Step 5.5.2.3

Rewrite the polynomial.

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

Step 5.5.2.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 1$ .

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