Calculus Examples

$f (x) = \frac{4}{x} + 2^{x - 6} - x + 5^{5}$

Step 1

Differentiate using the Sum Rule.

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Step 1.1

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

$\frac{d}{d x} [\frac{4}{x} + 2^{x - 6} - x + 3125]$

Step 1.2

By the Sum Rule, the derivative of $\frac{4}{x} + 2^{x - 6} - x + 3125$ with respect to $x$ is $\frac{d}{d x} [\frac{4}{x}] + \frac{d}{d x} [2^{x - 6}] + \frac{d}{d x} [- x] + \frac{d}{d x} [3125]$ .

$\frac{d}{d x} [\frac{4}{x}] + \frac{d}{d x} [2^{x - 6}] + \frac{d}{d x} [- x] + \frac{d}{d x} [3125]$

Step 2

Evaluate $\frac{d}{d x} [\frac{4}{x}]$ .

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Step 2.1

Since $4$ is constant with respect to $x$ , the derivative of $\frac{4}{x}$ with respect to $x$ is $4 \frac{d}{d x} [\frac{1}{x}]$ .

$4 \frac{d}{d x} [\frac{1}{x}] + \frac{d}{d x} [2^{x - 6}] + \frac{d}{d x} [- x] + \frac{d}{d x} [3125]$

Step 2.2

Rewrite $\frac{1}{x}$ as $x^{- 1}$ .

$4 \frac{d}{d x} [x^{- 1}] + \frac{d}{d x} [2^{x - 6}] + \frac{d}{d x} [- x] + \frac{d}{d x} [3125]$

Step 2.3

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

$4 (- x^{- 2}) + \frac{d}{d x} [2^{x - 6}] + \frac{d}{d x} [- x] + \frac{d}{d x} [3125]$

Step 2.4

Multiply $- 1$ by $4$ .

$- 4 x^{- 2} + \frac{d}{d x} [2^{x - 6}] + \frac{d}{d x} [- x] + \frac{d}{d x} [3125]$

Step 3

Evaluate $\frac{d}{d x} [2^{x - 6}]$ .

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Step 3.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = 2^{x}$ and $g (x) = x - 6$ .

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Step 3.1.1

To apply the Chain Rule, set $u$ as $x - 6$ .

$- 4 x^{- 2} + \frac{d}{d u} [2^{u}] \frac{d}{d x} [x - 6] + \frac{d}{d x} [- x] + \frac{d}{d x} [3125]$

Step 3.1.2

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

$- 4 x^{- 2} + 2^{u} \ln (2) \frac{d}{d x} [x - 6] + \frac{d}{d x} [- x] + \frac{d}{d x} [3125]$

Step 3.1.3

Replace all occurrences of $u$ with $x - 6$ .

$- 4 x^{- 2} + 2^{x - 6} \ln (2) \frac{d}{d x} [x - 6] + \frac{d}{d x} [- x] + \frac{d}{d x} [3125]$

Step 3.2

By the Sum Rule, the derivative of $x - 6$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 6]$ .

$- 4 x^{- 2} + 2^{x - 6} \ln (2) (\frac{d}{d x} [x] + \frac{d}{d x} [- 6]) + \frac{d}{d x} [- x] + \frac{d}{d x} [3125]$

Step 3.3

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

$- 4 x^{- 2} + 2^{x - 6} \ln (2) (1 + \frac{d}{d x} [- 6]) + \frac{d}{d x} [- x] + \frac{d}{d x} [3125]$

Step 3.4

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6$ with respect to $x$ is $0$ .

$- 4 x^{- 2} + 2^{x - 6} \ln (2) (1 + 0) + \frac{d}{d x} [- x] + \frac{d}{d x} [3125]$

Step 3.5

Add $1$ and $0$ .

$- 4 x^{- 2} + 2^{x - 6} \ln (2) \cdot 1 + \frac{d}{d x} [- x] + \frac{d}{d x} [3125]$

Step 3.6

Multiply $2^{x - 6}$ by $1$ .

$- 4 x^{- 2} + 2^{x - 6} \ln (2) + \frac{d}{d x} [- x] + \frac{d}{d x} [3125]$

Step 4

Evaluate $\frac{d}{d x} [- x]$ .

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Step 4.1

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

$- 4 x^{- 2} + 2^{x - 6} \ln (2) - \frac{d}{d x} [x] + \frac{d}{d x} [3125]$

Step 4.2

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

$- 4 x^{- 2} + 2^{x - 6} \ln (2) - 1 \cdot 1 + \frac{d}{d x} [3125]$

Step 4.3

Multiply $- 1$ by $1$ .

$- 4 x^{- 2} + 2^{x - 6} \ln (2) - 1 + \frac{d}{d x} [3125]$

Step 5

Since $3125$ is constant with respect to $x$ , the derivative of $3125$ with respect to $x$ is $0$ .

$- 4 x^{- 2} + 2^{x - 6} \ln (2) - 1 + 0$

Step 6

Simplify.

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Step 6.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- 4 \frac{1}{x^{2}} + 2^{x - 6} \ln (2) - 1 + 0$

Step 6.2

Combine terms.

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Step 6.2.1

Combine $- 4$ and $\frac{1}{x^{2}}$ .

$\frac{- 4}{x^{2}} + 2^{x - 6} \ln (2) - 1 + 0$

Step 6.2.2

Move the negative in front of the fraction.

$- \frac{4}{x^{2}} + 2^{x - 6} \ln (2) - 1 + 0$

Step 6.2.3

Add $- \frac{4}{x^{2}} + 2^{x - 6} \ln (2) - 1$ and $0$ .

$- \frac{4}{x^{2}} + 2^{x - 6} \ln (2) - 1$

Step 6.3

Reorder terms.

$2^{x - 6} \ln (2) - \frac{4}{x^{2}} - 1$