Calculus Examples

$\frac{e^{x}}{7 x^{2} + 7}$

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 $g (x) = 7 x^{2} + 7$ .

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

Step 2

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

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

Step 3

Differentiate.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

Multiply $2$ by $7$ .

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

Step 3.5

Since $7$ is constant with respect to $x$ , the derivative of $7$ with respect to $x$ is $0$ .

$\frac{(7 x^{2} + 7) e^{x} - e^{x} (14 x + 0)}{{(7 x^{2} + 7)}^{2}}$

Step 3.6

Simplify the expression.

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Step 3.6.1

Add $14 x$ and $0$ .

$\frac{(7 x^{2} + 7) e^{x} - e^{x} (14 x)}{{(7 x^{2} + 7)}^{2}}$

Step 3.6.2

Multiply $14$ by $- 1$ .

$\frac{(7 x^{2} + 7) e^{x} - 14 e^{x} x}{{(7 x^{2} + 7)}^{2}}$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$\frac{7 x^{2} e^{x} + 7 e^{x} - 14 e^{x} x}{{(7 x^{2} + 7)}^{2}}$

Step 4.2

Reorder factors in $7 x^{2} e^{x} + 7 e^{x} - 14 e^{x} x$ .

$\frac{7 x^{2} e^{x} + 7 e^{x} - 14 x e^{x}}{{(7 x^{2} + 7)}^{2}}$

Step 4.3

Reorder terms.

$\frac{7 e^{x} x^{2} - 14 e^{x} x + 7 e^{x}}{{(7 x^{2} + 7)}^{2}}$

Step 4.4

Simplify the numerator.

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Step 4.4.1

Factor $7 e^{x}$ out of $7 e^{x} x^{2} - 14 e^{x} x + 7 e^{x}$ .

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Step 4.4.1.1

Factor $7 e^{x}$ out of $7 e^{x} x^{2}$ .

$\frac{7 e^{x} (x^{2}) - 14 e^{x} x + 7 e^{x}}{{(7 x^{2} + 7)}^{2}}$

Step 4.4.1.2

Factor $7 e^{x}$ out of $- 14 e^{x} x$ .

$\frac{7 e^{x} (x^{2}) + 7 e^{x} (- 2 x) + 7 e^{x}}{{(7 x^{2} + 7)}^{2}}$

Step 4.4.1.3

Factor $7 e^{x}$ out of $7 e^{x}$ .

$\frac{7 e^{x} (x^{2}) + 7 e^{x} (- 2 x) + 7 e^{x} (1)}{{(7 x^{2} + 7)}^{2}}$

Step 4.4.1.4

Factor $7 e^{x}$ out of $7 e^{x} (x^{2}) + 7 e^{x} (- 2 x)$ .

$\frac{7 e^{x} (x^{2} - 2 x) + 7 e^{x} (1)}{{(7 x^{2} + 7)}^{2}}$

Step 4.4.1.5

Factor $7 e^{x}$ out of $7 e^{x} (x^{2} - 2 x) + 7 e^{x} (1)$ .

$\frac{7 e^{x} (x^{2} - 2 x + 1)}{{(7 x^{2} + 7)}^{2}}$

Step 4.4.2

Factor using the perfect square rule.

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Step 4.4.2.1

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

$\frac{7 e^{x} (x^{2} - 2 x + 1^{2})}{{(7 x^{2} + 7)}^{2}}$

Step 4.4.2.2

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 4.4.2.3

Rewrite the polynomial.

$\frac{7 e^{x} (x^{2} - 2 \cdot x \cdot 1 + 1^{2})}{{(7 x^{2} + 7)}^{2}}$

Step 4.4.2.4

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

$\frac{7 e^{x} {(x - 1)}^{2}}{{(7 x^{2} + 7)}^{2}}$

Step 4.5

Simplify the denominator.

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Step 4.5.1

Factor $7$ out of $7 x^{2} + 7$ .

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Step 4.5.1.1

Factor $7$ out of $7 x^{2}$ .

$\frac{7 e^{x} {(x - 1)}^{2}}{{(7 (x^{2}) + 7)}^{2}}$

Step 4.5.1.2

Factor $7$ out of $7$ .

$\frac{7 e^{x} {(x - 1)}^{2}}{{(7 (x^{2}) + 7 (1))}^{2}}$

Step 4.5.1.3

Factor $7$ out of $7 (x^{2}) + 7 (1)$ .

$\frac{7 e^{x} {(x - 1)}^{2}}{{(7 (x^{2} + 1))}^{2}}$

Step 4.5.2

Apply the product rule to $7 (x^{2} + 1)$ .

$\frac{7 e^{x} {(x - 1)}^{2}}{7^{2} {(x^{2} + 1)}^{2}}$

Step 4.5.3

Raise $7$ to the power of $2$ .

$\frac{7 e^{x} {(x - 1)}^{2}}{49 {(x^{2} + 1)}^{2}}$

Step 4.6

Cancel the common factor of $7$ and $49$ .

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Step 4.6.1

Factor $7$ out of $7 e^{x} {(x - 1)}^{2}$ .

$\frac{7 (e^{x} {(x - 1)}^{2})}{49 {(x^{2} + 1)}^{2}}$

Step 4.6.2

Cancel the common factors.

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Step 4.6.2.1

Factor $7$ out of $49 {(x^{2} + 1)}^{2}$ .

$\frac{7 (e^{x} {(x - 1)}^{2})}{7 (7 {(x^{2} + 1)}^{2})}$

Step 4.6.2.2

Cancel the common factor.

$\frac{7 (e^{x} {(x - 1)}^{2})}{7 (7 {(x^{2} + 1)}^{2})}$

Step 4.6.2.3

Rewrite the expression.

$\frac{e^{x} {(x - 1)}^{2}}{7 {(x^{2} + 1)}^{2}}$