Calculus Examples

$\frac{5 x}{2^{x}}$

Step 1

Since $5$ is constant with respect to $x$ , the derivative of $\frac{5 x}{2^{x}}$ with respect to $x$ is $5 \frac{d}{d x} [\frac{x}{2^{x}}]$ .

$5 \frac{d}{d x} [\frac{x}{2^{x}}]$

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

$5 \frac{2^{x} \frac{d}{d x} [x] - x \frac{d}{d x} [2^{x}]}{{(2^{x})}^{2}}$

Step 3

Differentiate using the Power Rule.

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Step 3.1

Multiply the exponents in ${(2^{x})}^{2}$ .

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Step 3.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$5 \frac{2^{x} \frac{d}{d x} [x] - x \frac{d}{d x} [2^{x}]}{2^{x \cdot 2}}$

Step 3.1.2

Move $2$ to the left of $x$ .

$5 \frac{2^{x} \frac{d}{d x} [x] - x \frac{d}{d x} [2^{x}]}{2^{2 x}}$

Step 3.2

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

$5 \frac{2^{x} \cdot 1 - x \frac{d}{d x} [2^{x}]}{2^{2 x}}$

Step 3.3

Multiply $2^{x}$ by $1$ .

$5 \frac{2^{x} - x \frac{d}{d x} [2^{x}]}{2^{2 x}}$

Step 4

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

$5 \frac{2^{x} - x (2^{x} \ln (2))}{2^{2 x}}$

Step 5

Combine $5$ and $\frac{2^{x} - x (2^{x} \ln (2))}{2^{2 x}}$ .

$\frac{5 (2^{x} - x (2^{x} \ln (2)))}{2^{2 x}}$

Step 6

Simplify.

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Step 6.1

Apply the distributive property.

$\frac{5 \cdot 2^{x} + 5 (- x (2^{x} \ln (2)))}{2^{2 x}}$

Step 6.2

Simplify the numerator.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

Multiply $5 (- x (2^{x} \ln (2)))$ .

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Step 6.2.1.1.1

Multiply $- 1$ by $5$ .

$\frac{5 \cdot 2^{x} - 5 (x (2^{x} \ln (2)))}{2^{2 x}}$

Step 6.2.1.1.2

Simplify $- 5 \ln (2)$ by moving $5$ inside the logarithm.

$\frac{5 \cdot 2^{x} + x (2^{x} (- \ln (2^{5})))}{2^{2 x}}$

Step 6.2.1.2

Rewrite using the commutative property of multiplication.

$\frac{5 \cdot 2^{x} - \ln (2^{5}) x \cdot 2^{x}}{2^{2 x}}$

Step 6.2.1.3

Raise $2$ to the power of $5$ .

$\frac{5 \cdot 2^{x} - \ln (32) x \cdot 2^{x}}{2^{2 x}}$

Step 6.2.2

Reorder factors in $5 \cdot 2^{x} - \ln (32) x \cdot 2^{x}$ .

$\frac{5 \cdot 2^{x} - x \cdot 2^{x} \ln (32)}{2^{2 x}}$

Step 6.3

Reorder terms.

$\frac{- 2^{x} x \ln (32) + 5 \cdot 2^{x}}{2^{2 x}}$

Step 6.4

Factor $2^{x}$ out of $- 2^{x} x \ln (32) + 5 \cdot 2^{x}$ .

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Step 6.4.1

Factor $2^{x}$ out of $- 2^{x} x \ln (32)$ .

$\frac{2^{x} (- x \ln (32)) + 5 \cdot 2^{x}}{2^{2 x}}$

Step 6.4.2

Factor $2^{x}$ out of $5 \cdot 2^{x}$ .

$\frac{2^{x} (- x \ln (32)) + 2^{x} \cdot 5}{2^{2 x}}$

Step 6.4.3

Factor $2^{x}$ out of $2^{x} (- x \ln (32)) + 2^{x} \cdot 5$ .

$\frac{2^{x} (- x \ln (32) + 5)}{2^{2 x}}$

Step 6.5

Cancel the common factor of $2^{x}$ and $2^{2 x}$ .

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Step 6.5.1

Factor $2^{2 x}$ out of $2^{x} (- x \ln (32) + 5)$ .

$\frac{2^{2 x} (2^{- x} (- x \ln (32) + 5))}{2^{2 x}}$

Step 6.5.2

Cancel the common factors.

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Step 6.5.2.1

Multiply by $1$ .

$\frac{2^{2 x} (2^{- x} (- x \ln (32) + 5))}{2^{2 x} \cdot 1}$

Step 6.5.2.2

Cancel the common factor.

$\frac{2^{2 x} (2^{- x} (- x \ln (32) + 5))}{2^{2 x} \cdot 1}$

Step 6.5.2.3

Rewrite the expression.

$\frac{2^{- x} (- x \ln (32) + 5)}{1}$

Step 6.5.2.4

Divide $2^{- x} (- x \ln (32) + 5)$ by $1$ .

$2^{- x} (- x \ln (32) + 5)$

Step 6.6

Apply the distributive property.

$2^{- x} (- x \ln (32)) + 2^{- x} \cdot 5$

Step 6.7

Rewrite using the commutative property of multiplication.

$- \ln (32) \cdot 2^{- x} x + 2^{- x} \cdot 5$

Step 6.8

Move $5$ to the left of $2^{- x}$ .

$- \ln (32) \cdot 2^{- x} x + 5 \cdot 2^{- x}$

Step 6.9

Reorder factors in $- \ln (32) \cdot 2^{- x} x + 5 \cdot 2^{- x}$ .

$- x \cdot 2^{- x} \ln (32) + 5 \cdot 2^{- x}$