Calculus Examples

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

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 $g (x) = 2 x + 1$ .

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

Step 2

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

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

Step 3

Differentiate.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

Multiply $2$ by $1$ .

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

Step 3.5

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

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

Step 3.6

Simplify the expression.

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Step 3.6.1

Add $2$ and $0$ .

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

Step 3.6.2

Multiply $2$ by $- 1$ .

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

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Simplify the numerator.

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Step 4.2.1

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

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

Step 4.2.2

Subtract $2 e^{x}$ from $e^{x}$ .

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

Step 4.3

Reorder terms.

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

Step 4.4

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

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Step 4.4.1

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

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

Step 4.4.2

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

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

Step 4.4.3

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

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