Calculus Examples

Let $h (x) = \frac{e^{2 x}}{x - 3}$

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 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 $f (x) = e^{2 x}$ and $g (x) = x - 3$ .

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

Step 1.1.2

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 $g (x) = 2 x$ .

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Step 1.1.2.1

To apply the Chain Rule, set $u$ as $2 x$ .

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

Step 1.1.2.2

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

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

Step 1.1.2.3

Replace all occurrences of $u$ with $2 x$ .

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

Step 1.1.3

Differentiate.

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Step 1.1.3.1

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

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

Step 1.1.3.2

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

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

Step 1.1.3.3

Simplify the expression.

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Step 1.1.3.3.1

Multiply $2$ by $1$ .

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

Step 1.1.3.3.2

Move $2$ to the left of $(x - 3) e^{2 x}$ .

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

Step 1.1.3.4

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

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

Step 1.1.3.5

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

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

Step 1.1.3.6

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

$\frac{2 (x - 3) e^{2 x} - e^{2 x} (1 + 0)}{{(x - 3)}^{2}}$

Step 1.1.3.7

Simplify the expression.

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Step 1.1.3.7.1

Add $1$ and $0$ .

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

Step 1.1.3.7.2

Multiply $- 1$ by $1$ .

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

Step 1.1.4

Simplify.

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Step 1.1.4.1

Apply the distributive property.

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

Step 1.1.4.2

Apply the distributive property.

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

Step 1.1.4.3

Simplify the numerator.

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Step 1.1.4.3.1

Multiply $2$ by $- 3$ .

$\frac{2 x e^{2 x} - 6 e^{2 x} - e^{2 x}}{{(x - 3)}^{2}}$

Step 1.1.4.3.2

Subtract $e^{2 x}$ from $- 6 e^{2 x}$ .

$\frac{2 x e^{2 x} - 7 e^{2 x}}{{(x - 3)}^{2}}$

Step 1.1.4.4

Reorder terms.

$\frac{2 e^{2 x} x - 7 e^{2 x}}{{(x - 3)}^{2}}$

Step 1.1.4.5

Factor $e^{2 x}$ out of $2 e^{2 x} x - 7 e^{2 x}$ .

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Step 1.1.4.5.1

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

$\frac{e^{2 x} (2 x) - 7 e^{2 x}}{{(x - 3)}^{2}}$

Step 1.1.4.5.2

Factor $e^{2 x}$ out of $- 7 e^{2 x}$ .

$\frac{e^{2 x} (2 x) + e^{2 x} \cdot - 7}{{(x - 3)}^{2}}$

Step 1.1.4.5.3

Factor $e^{2 x}$ out of $e^{2 x} (2 x) + e^{2 x} \cdot - 7$ .

$f' (x) = \frac{e^{2 x} (2 x - 7)}{{(x - 3)}^{2}}$

Step 1.2

The first derivative of $h (x)$ with respect to $x$ is $\frac{e^{2 x} (2 x - 7)}{{(x - 3)}^{2}}$ .

$\frac{e^{2 x} (2 x - 7)}{{(x - 3)}^{2}}$

Step 2

Set the first derivative equal to $0$ then solve the equation $\frac{e^{2 x} (2 x - 7)}{{(x - 3)}^{2}} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$\frac{e^{2 x} (2 x - 7)}{{(x - 3)}^{2}} = 0$

Step 2.2

Set the numerator equal to zero.

$e^{2 x} (2 x - 7) = 0$

Step 2.3

Solve the equation for $x$ .

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Step 2.3.1

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$e^{2 x} = 0$

$2 x - 7 = 0$

Step 2.3.2

Set $e^{2 x}$ equal to $0$ and solve for $x$ .

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Step 2.3.2.1

Set $e^{2 x}$ equal to $0$ .

$e^{2 x} = 0$

Step 2.3.2.2

Solve $e^{2 x} = 0$ for $x$ .

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Step 2.3.2.2.1

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (e^{2 x}) = \ln (0)$

Step 2.3.2.2.2

The equation cannot be solved because $\ln (0)$ is undefined.

Undefined

Step 2.3.2.2.3

There is no solution for $e^{2 x} = 0$

No solution

Step 2.3.3

Set $2 x - 7$ equal to $0$ and solve for $x$ .

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Step 2.3.3.1

Set $2 x - 7$ equal to $0$ .

$2 x - 7 = 0$

Step 2.3.3.2

Solve $2 x - 7 = 0$ for $x$ .

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Step 2.3.3.2.1

Add $7$ to both sides of the equation.

$2 x = 7$

Step 2.3.3.2.2

Divide each term in $2 x = 7$ by $2$ and simplify.

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Step 2.3.3.2.2.1

Divide each term in $2 x = 7$ by $2$ .

$\frac{2 x}{2} = \frac{7}{2}$

Step 2.3.3.2.2.2

Simplify the left side.

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Step 2.3.3.2.2.2.1

Cancel the common factor of $2$ .

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Step 2.3.3.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{7}{2}$

Step 2.3.3.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{7}{2}$

Step 2.3.4

The final solution is all the values that make $e^{2 x} (2 x - 7) = 0$ true.

$x = \frac{7}{2}$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

Set the denominator in $\frac{e^{2 x} (2 x - 7)}{{(x - 3)}^{2}}$ equal to $0$ to find where the expression is undefined.

${(x - 3)}^{2} = 0$

Step 3.2

Solve for $x$ .

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Step 3.2.1

Set the $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 3.2.2

Add $3$ to both sides of the equation.

$x = 3$

Step 4

Evaluate $\frac{e^{2 x}}{x - 3}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = \frac{7}{2}$ .

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Step 4.1.1

Substitute $\frac{7}{2}$ for $x$ .

$\frac{e^{2 (\frac{7}{2})}}{(\frac{7}{2}) - 3}$

Step 4.1.2

Simplify.

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Step 4.1.2.1

Cancel the common factor of $2$ .

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Step 4.1.2.1.1

Cancel the common factor.

$\frac{e^{2 (\frac{7}{2})}}{\frac{7}{2} - 3}$

Step 4.1.2.1.2

Rewrite the expression.

$\frac{e^{7}}{\frac{7}{2} - 3}$

Step 4.1.2.2

Simplify the denominator.

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Step 4.1.2.2.1

To write $- 3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{e^{7}}{\frac{7}{2} - 3 \cdot \frac{2}{2}}$

Step 4.1.2.2.2

Combine $- 3$ and $\frac{2}{2}$ .

$\frac{e^{7}}{\frac{7}{2} + \frac{- 3 \cdot 2}{2}}$

Step 4.1.2.2.3

Combine the numerators over the common denominator.

$\frac{e^{7}}{\frac{7 - 3 \cdot 2}{2}}$

Step 4.1.2.2.4

Simplify the numerator.

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Step 4.1.2.2.4.1

Multiply $- 3$ by $2$ .

$\frac{e^{7}}{\frac{7 - 6}{2}}$

Step 4.1.2.2.4.2

Subtract $6$ from $7$ .

$\frac{e^{7}}{\frac{1}{2}}$

Step 4.1.2.3

Multiply the numerator by the reciprocal of the denominator.

$e^{7} \cdot 2$

Step 4.1.2.4

Move $2$ to the left of $e^{7}$ .

$2 e^{7}$

Step 4.2

Evaluate at $x = 3$ .

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Step 4.2.1

Substitute $3$ for $x$ .

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

Step 4.2.2

Simplify.

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Step 4.2.2.1

Subtract $3$ from $3$ .

$\frac{e^{2 (3)}}{0}$

Step 4.2.2.2

The expression contains a division by $0$ . The expression is undefined.

Undefined

Step 4.3

List all of the points.

$(\frac{7}{2}, 2 e^{7})$

Step 5