Calculus Examples

$f (x) = \frac{2^{x}}{x}$

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 $f (x) = 2^{x}$ and $g (x) = x$ .

$\frac{x \frac{d}{d x} [2^{x}] - 2^{x} \frac{d}{d x} [x]}{x^{2}}$

Step 2

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $2$ .

$\frac{x (2^{x} \ln (2)) - 2^{x} \frac{d}{d x} [x]}{x^{2}}$

Step 3

Differentiate using the Power Rule.

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Step 3.1

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

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

Step 3.2

Simplify the expression.

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Step 3.2.1

Multiply $- 1$ by $1$ .

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

Step 3.2.2

Rewrite $- 1 \cdot 2^{x}$ as $- 2^{x}$ .

$\frac{x (2^{x} \ln (2)) - 2^{x}}{x^{2}}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Rewrite using the commutative property of multiplication.

$\frac{\ln (2) x \cdot 2^{x} - 2^{x}}{x^{2}}$

Step 4.1.2

Reorder factors in $\ln (2) x \cdot 2^{x} - 2^{x}$ .

$\frac{x \cdot 2^{x} \ln (2) - 2^{x}}{x^{2}}$

Step 4.2

Reorder terms.

$\frac{2^{x} x \ln (2) - 2^{x}}{x^{2}}$

Step 4.3

Factor $2^{x}$ out of $2^{x} x \ln (2) - 2^{x}$ .

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Step 4.3.1

Factor $2^{x}$ out of $2^{x} x \ln (2)$ .

$\frac{2^{x} (x \ln (2)) - 2^{x}}{x^{2}}$

Step 4.3.2

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

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

Step 4.3.3

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

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