Calculus Examples

$f (x) = \frac{e^{x} + e^{- x}}{8}$

Step 1

Find the first derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

Differentiate.

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Step 1.1.1.1

Since $\frac{1}{8}$ is constant with respect to $x$ , the derivative of $\frac{e^{x} + e^{- x}}{8}$ with respect to $x$ is $\frac{1}{8} \frac{d}{d x} [e^{x} + e^{- x}]$ .

$\frac{1}{8} \frac{d}{d x} [e^{x} + e^{- x}]$

Step 1.1.1.2

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

$\frac{1}{8} (\frac{d}{d x} [e^{x}] + \frac{d}{d x} [e^{- x}])$

Step 1.1.2

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

$\frac{1}{8} (e^{x} + \frac{d}{d x} [e^{- x}])$

Step 1.1.3

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

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Step 1.1.3.1

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

$\frac{1}{8} (e^{x} + \frac{d}{d u} [e^{u}] \frac{d}{d x} [- x])$

Step 1.1.3.2

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

$\frac{1}{8} (e^{x} + e^{u} \frac{d}{d x} [- x])$

Step 1.1.3.3

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

$\frac{1}{8} (e^{x} + e^{- x} \frac{d}{d x} [- x])$

Step 1.1.4

Differentiate.

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Step 1.1.4.1

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

$\frac{1}{8} (e^{x} + e^{- x} (- \frac{d}{d x} [x]))$

Step 1.1.4.2

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

$\frac{1}{8} (e^{x} + e^{- x} (- 1 \cdot 1))$

Step 1.1.4.3

Simplify the expression.

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Step 1.1.4.3.1

Multiply $- 1$ by $1$ .

$\frac{1}{8} (e^{x} + e^{- x} \cdot - 1)$

Step 1.1.4.3.2

Move $- 1$ to the left of $e^{- x}$ .

$\frac{1}{8} (e^{x} - 1 \cdot e^{- x})$

Step 1.1.4.3.3

Rewrite $- 1 e^{- x}$ as $- e^{- x}$ .

$\frac{1}{8} (e^{x} - e^{- x})$

Step 1.1.5

Simplify.

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Step 1.1.5.1

Apply the distributive property.

$\frac{1}{8} e^{x} + \frac{1}{8} (- e^{- x})$

Step 1.1.5.2

Combine $\frac{1}{8}$ and $e^{- x}$ .

$\frac{1}{8} e^{x} - \frac{e^{- x}}{8}$

Step 1.1.5.3

Combine $\frac{1}{8}$ and $e^{x}$ .

$f' (x) = \frac{e^{x}}{8} - \frac{e^{- x}}{8}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{e^{x}}{8} - \frac{e^{- x}}{8}$ .

$\frac{e^{x}}{8} - \frac{e^{- x}}{8}$

Step 2

Set the first derivative equal to $0$ then solve the equation $\frac{e^{x}}{8} - \frac{e^{- x}}{8} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$\frac{e^{x}}{8} - \frac{e^{- x}}{8} = 0$

Step 2.2

Move $- \frac{e^{- x}}{8}$ to the right side of the equation by adding it to both sides.

$\frac{e^{x}}{8} = \frac{e^{- x}}{8}$

Step 2.3

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$e^{x} = e^{- x}$

Step 2.4

Since the bases are the same, then two expressions are only equal if the exponents are also equal.

$x = - x$

Step 2.5

Solve for $x$ .

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Step 2.5.1

Move all terms containing $x$ to the left side of the equation.

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Step 2.5.1.1

Add $x$ to both sides of the equation.

$x + x = 0$

Step 2.5.1.2

Add $x$ and $x$ .

$2 x = 0$

Step 2.5.2

Divide each term in $2 x = 0$ by $2$ and simplify.

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Step 2.5.2.1

Divide each term in $2 x = 0$ by $2$ .

$\frac{2 x}{2} = \frac{0}{2}$

Step 2.5.2.2

Simplify the left side.

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Step 2.5.2.2.1

Cancel the common factor of $2$ .

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Step 2.5.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{0}{2}$

Step 2.5.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{2}$

Step 2.5.2.3

Simplify the right side.

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Step 2.5.2.3.1

Divide $0$ by $2$ .

$x = 0$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 4

Evaluate $\frac{e^{x} + e^{- x}}{8}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = 0$ .

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Step 4.1.1

Substitute $0$ for $x$ .

$\frac{e^{0} + e^{- (0)}}{8}$

Step 4.1.2

Simplify.

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Step 4.1.2.1

Simplify the numerator.

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Step 4.1.2.1.1

Anything raised to $0$ is $1$ .

$\frac{1 + e^{- (0)}}{8}$

Step 4.1.2.1.2

Multiply $- 1$ by $0$ .

$\frac{1 + e^{0}}{8}$

Step 4.1.2.1.3

Anything raised to $0$ is $1$ .

$\frac{1 + 1}{8}$

Step 4.1.2.1.4

Add $1$ and $1$ .

$\frac{2}{8}$

Step 4.1.2.2

Cancel the common factor of $2$ and $8$ .

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Step 4.1.2.2.1

Factor $2$ out of $2$ .

$\frac{2 (1)}{8}$

Step 4.1.2.2.2

Cancel the common factors.

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Step 4.1.2.2.2.1

Factor $2$ out of $8$ .

$\frac{2 \cdot 1}{2 \cdot 4}$

Step 4.1.2.2.2.2

Cancel the common factor.

$\frac{2 \cdot 1}{2 \cdot 4}$

Step 4.1.2.2.2.3

Rewrite the expression.

$\frac{1}{4}$

Step 4.2

List all of the points.

$(0, \frac{1}{4})$

Step 5