Calculus Examples

$y = \frac{e^{3 x} + e^{- 3 x}}{x}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\frac{e^{3 x} + e^{- 3 x}}{x})$

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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 $f (x) = e^{3 x} + e^{- 3 x}$ and $g (x) = x$ .

$\frac{x \frac{d}{d x} [e^{3 x} + e^{- 3 x}] - (e^{3 x} + e^{- 3 x}) \frac{d}{d x} [x]}{x^{2}}$

Step 3.2

By the Sum Rule, the derivative of $e^{3 x} + e^{- 3 x}$ with respect to $x$ is $\frac{d}{d x} [e^{3 x}] + \frac{d}{d x} [e^{- 3 x}]$ .

$\frac{x (\frac{d}{d x} [e^{3 x}] + \frac{d}{d x} [e^{- 3 x}]) - (e^{3 x} + e^{- 3 x}) \frac{d}{d x} [x]}{x^{2}}$

Step 3.3

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 $g (x) = 3 x$ .

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Step 3.3.1

To apply the Chain Rule, set $u_{1}$ as $3 x$ .

$\frac{x (\frac{d}{d u_{1}} [e^{u_{1}}] \frac{d}{d x} [3 x] + \frac{d}{d x} [e^{- 3 x}]) - (e^{3 x} + e^{- 3 x}) \frac{d}{d x} [x]}{x^{2}}$

Step 3.3.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{1}} [a^{u_{1}}]$ is $a^{u_{1}} \ln (a)$ where $a$ = $e$ .

$\frac{x (e^{u_{1}} \frac{d}{d x} [3 x] + \frac{d}{d x} [e^{- 3 x}]) - (e^{3 x} + e^{- 3 x}) \frac{d}{d x} [x]}{x^{2}}$

Step 3.3.3

Replace all occurrences of $u_{1}$ with $3 x$ .

$\frac{x (e^{3 x} \frac{d}{d x} [3 x] + \frac{d}{d x} [e^{- 3 x}]) - (e^{3 x} + e^{- 3 x}) \frac{d}{d x} [x]}{x^{2}}$

Step 3.4

Differentiate.

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Step 3.4.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$\frac{x (e^{3 x} (3 \frac{d}{d x} [x]) + \frac{d}{d x} [e^{- 3 x}]) - (e^{3 x} + e^{- 3 x}) \frac{d}{d x} [x]}{x^{2}}$

Step 3.4.2

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

$\frac{x (e^{3 x} (3 \cdot 1) + \frac{d}{d x} [e^{- 3 x}]) - (e^{3 x} + e^{- 3 x}) \frac{d}{d x} [x]}{x^{2}}$

Step 3.4.3

Simplify the expression.

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Step 3.4.3.1

Multiply $3$ by $1$ .

$\frac{x (e^{3 x} \cdot 3 + \frac{d}{d x} [e^{- 3 x}]) - (e^{3 x} + e^{- 3 x}) \frac{d}{d x} [x]}{x^{2}}$

Step 3.4.3.2

Move $3$ to the left of $e^{3 x}$ .

$\frac{x (3 \cdot e^{3 x} + \frac{d}{d x} [e^{- 3 x}]) - (e^{3 x} + e^{- 3 x}) \frac{d}{d x} [x]}{x^{2}}$

Step 3.5

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 $g (x) = - 3 x$ .

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Step 3.5.1

To apply the Chain Rule, set $u_{2}$ as $- 3 x$ .

$\frac{x (3 e^{3 x} + \frac{d}{d u_{2}} [e^{u_{2}}] \frac{d}{d x} [- 3 x]) - (e^{3 x} + e^{- 3 x}) \frac{d}{d x} [x]}{x^{2}}$

Step 3.5.2

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

$\frac{x (3 e^{3 x} + e^{u_{2}} \frac{d}{d x} [- 3 x]) - (e^{3 x} + e^{- 3 x}) \frac{d}{d x} [x]}{x^{2}}$

Step 3.5.3

Replace all occurrences of $u_{2}$ with $- 3 x$ .

$\frac{x (3 e^{3 x} + e^{- 3 x} \frac{d}{d x} [- 3 x]) - (e^{3 x} + e^{- 3 x}) \frac{d}{d x} [x]}{x^{2}}$

Step 3.6

Differentiate.

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Step 3.6.1

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 x$ with respect to $x$ is $- 3 \frac{d}{d x} [x]$ .

$\frac{x (3 e^{3 x} + e^{- 3 x} (- 3 \frac{d}{d x} [x])) - (e^{3 x} + e^{- 3 x}) \frac{d}{d x} [x]}{x^{2}}$

Step 3.6.2

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

$\frac{x (3 e^{3 x} + e^{- 3 x} (- 3 \cdot 1)) - (e^{3 x} + e^{- 3 x}) \frac{d}{d x} [x]}{x^{2}}$

Step 3.6.3

Simplify the expression.

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Step 3.6.3.1

Multiply $- 3$ by $1$ .

$\frac{x (3 e^{3 x} + e^{- 3 x} \cdot - 3) - (e^{3 x} + e^{- 3 x}) \frac{d}{d x} [x]}{x^{2}}$

Step 3.6.3.2

Move $- 3$ to the left of $e^{- 3 x}$ .

$\frac{x (3 e^{3 x} - 3 \cdot e^{- 3 x}) - (e^{3 x} + e^{- 3 x}) \frac{d}{d x} [x]}{x^{2}}$

Step 3.6.4

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

$\frac{x (3 e^{3 x} - 3 e^{- 3 x}) - (e^{3 x} + e^{- 3 x}) \cdot 1}{x^{2}}$

Step 3.6.5

Multiply $- 1$ by $1$ .

$\frac{x (3 e^{3 x} - 3 e^{- 3 x}) - (e^{3 x} + e^{- 3 x})}{x^{2}}$

Step 3.7

Simplify.

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Step 3.7.1

Apply the distributive property.

$\frac{x (3 e^{3 x}) + x (- 3 e^{- 3 x}) - (e^{3 x} + e^{- 3 x})}{x^{2}}$

Step 3.7.2

Apply the distributive property.

$\frac{x (3 e^{3 x}) + x (- 3 e^{- 3 x}) - e^{3 x} - e^{- 3 x}}{x^{2}}$

Step 3.7.3

Simplify each term.

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Step 3.7.3.1

Rewrite using the commutative property of multiplication.

$\frac{3 x e^{3 x} + x (- 3 e^{- 3 x}) - e^{3 x} - e^{- 3 x}}{x^{2}}$

Step 3.7.3.2

Rewrite using the commutative property of multiplication.

$\frac{3 x e^{3 x} - 3 x e^{- 3 x} - e^{3 x} - e^{- 3 x}}{x^{2}}$

Step 3.7.4

Reorder terms.

$\frac{3 e^{3 x} x - 3 e^{- 3 x} x - e^{3 x} - e^{- 3 x}}{x^{2}}$

Step 3.7.5

Reorder factors in $\frac{3 e^{3 x} x - 3 e^{- 3 x} x - e^{3 x} - e^{- 3 x}}{x^{2}}$ .

$\frac{3 x e^{3 x} - 3 x e^{- 3 x} - e^{3 x} - e^{- 3 x}}{x^{2}}$

Step 4

Reform the equation by setting the left side equal to the right side.

$y' = \frac{3 x e^{3 x} - 3 x e^{- 3 x} - e^{3 x} - e^{- 3 x}}{x^{2}}$

Step 5

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = \frac{3 x e^{3 x} - 3 x e^{- 3 x} - e^{3 x} - e^{- 3 x}}{x^{2}}$