Calculus Examples

$e^{x} \sin (x)$

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 Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = e^{x}$ and $g (x) = \sin (x)$ .

$f' (x) = e^{x} \frac{d}{d x} (\sin (x)) + \sin (x) \frac{d}{d x} (e^{x})$

Step 1.1.2

The derivative of $\sin (x)$ with respect to $x$ is $\cos (x)$ .

$f' (x) = e^{x} \cos (x) + \sin (x) \frac{d}{d x} (e^{x})$

Step 1.1.3

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

$f' (x) = e^{x} \cos (x) + \sin (x) e^{x}$

Step 1.1.4

Reorder terms.

$f' (x) = e^{x} \cos (x) + e^{x} \sin (x)$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $e^{x} \cos (x) + e^{x} \sin (x)$ .

$e^{x} \cos (x) + e^{x} \sin (x)$

Step 2

Set the first derivative equal to $0$ then solve the equation $e^{x} \cos (x) + e^{x} \sin (x) = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$e^{x} \cos (x) + e^{x} \sin (x) = 0$

Step 2.2

Factor $e^{x}$ out of $e^{x} \cos (x) + e^{x} \sin (x)$ .

$e^{x} (\cos (x) + \sin (x)) = 0$

Step 2.3

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

$e^{x} = 0$

$\cos (x) + \sin (x) = 0$

Step 2.4

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

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Step 2.4.1

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

$e^{x} = 0$

Step 2.4.2

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

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Step 2.4.2.1

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

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

Step 2.4.2.2

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

Undefined

Step 2.4.2.3

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

No solution

Step 2.5

Set $\cos (x) + \sin (x)$ equal to $0$ and solve for $x$ .

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Step 2.5.1

Set $\cos (x) + \sin (x)$ equal to $0$ .

$\cos (x) + \sin (x) = 0$

Step 2.5.2

Solve $\cos (x) + \sin (x) = 0$ for $x$ .

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Step 2.5.2.1

Divide each term in the equation by $\cos (x)$ .

$\frac{\cos (x)}{\cos (x)} + \frac{\sin (x)}{\cos (x)} = \frac{0}{\cos (x)}$

Step 2.5.2.2

Cancel the common factor of $\cos (x)$ .

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Step 2.5.2.2.1

Cancel the common factor.

$\frac{\cos (x)}{\cos (x)} + \frac{\sin (x)}{\cos (x)} = \frac{0}{\cos (x)}$

Step 2.5.2.2.2

Rewrite the expression.

$1 + \frac{\sin (x)}{\cos (x)} = \frac{0}{\cos (x)}$

Step 2.5.2.3

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

$1 + \tan (x) = \frac{0}{\cos (x)}$

Step 2.5.2.4

Separate fractions.

$1 + \tan (x) = \frac{0}{1} \cdot \frac{1}{\cos (x)}$

Step 2.5.2.5

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$1 + \tan (x) = \frac{0}{1} \cdot \sec (x)$

Step 2.5.2.6

Divide $0$ by $1$ .

$1 + \tan (x) = 0 \sec (x)$

Step 2.5.2.7

Multiply $0$ by $\sec (x)$ .

$1 + \tan (x) = 0$

Step 2.5.2.8

Subtract $1$ from both sides of the equation.

$\tan (x) = - 1$

Step 2.5.2.9

Take the inverse tangent of both sides of the equation to extract $x$ from inside the tangent.

$x = \arctan (- 1)$

Step 2.5.2.10

Simplify the right side.

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Step 2.5.2.10.1

The exact value of $\arctan (- 1)$ is $- \frac{π}{4}$ .

$x = - \frac{π}{4}$

Step 2.5.2.11

The tangent function is negative in the second and fourth quadrants. To find the second solution, subtract the reference angle from $π$ to find the solution in the third quadrant.

$x = - \frac{π}{4} - π$

Step 2.5.2.12

Simplify the expression to find the second solution.

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Step 2.5.2.12.1

Add $2 π$ to $- \frac{π}{4} - π$ .

$x = - \frac{π}{4} - π + 2 π$

Step 2.5.2.12.2

The resulting angle of $\frac{3 π}{4}$ is positive and coterminal with $- \frac{π}{4} - π$ .

$x = \frac{3 π}{4}$

Step 2.5.2.13

Find the period of $\tan (x)$ .

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Step 2.5.2.13.1

The period of the function can be calculated using $\frac{π}{| b |}$ .

$\frac{π}{| b |}$

Step 2.5.2.13.2

Replace $b$ with $1$ in the formula for period.

$\frac{π}{| 1 |}$

Step 2.5.2.13.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{π}{1}$

Step 2.5.2.13.4

Divide $π$ by $1$ .

$π$

Step 2.5.2.14

Add $π$ to every negative angle to get positive angles.

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Step 2.5.2.14.1

Add $π$ to $- \frac{π}{4}$ to find the positive angle.

$- \frac{π}{4} + π$

Step 2.5.2.14.2

To write $π$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$π \cdot \frac{4}{4} - \frac{π}{4}$

Step 2.5.2.14.3

Combine fractions.

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Step 2.5.2.14.3.1

Combine $π$ and $\frac{4}{4}$ .

$\frac{π \cdot 4}{4} - \frac{π}{4}$

Step 2.5.2.14.3.2

Combine the numerators over the common denominator.

$\frac{π \cdot 4 - π}{4}$

Step 2.5.2.14.4

Simplify the numerator.

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Step 2.5.2.14.4.1

Move $4$ to the left of $π$ .

$\frac{4 \cdot π - π}{4}$

Step 2.5.2.14.4.2

Subtract $π$ from $4 π$ .

$\frac{3 π}{4}$

Step 2.5.2.14.5

List the new angles.

$x = \frac{3 π}{4}$

Step 2.5.2.15

The period of the $\tan (x)$ function is $π$ so values will repeat every $π$ radians in both directions.

$x = \frac{3 π}{4} + π n, \frac{3 π}{4} + π n$ , for any integer $n$

Step 2.6

The final solution is all the values that make $e^{x} (\cos (x) + \sin (x)) = 0$ true.

$x = \frac{3 π}{4} + π n$ , for any integer $n$

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 $e^{x} \sin (x)$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = \frac{3 π}{4}$ .

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Step 4.1.1

Substitute $\frac{3 π}{4}$ for $x$ .

$e^{\frac{3 π}{4}} \sin (\frac{3 π}{4})$

Step 4.1.2

Simplify.

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Step 4.1.2.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$e^{\frac{3 π}{4}} \sin (\frac{π}{4})$

Step 4.1.2.2

The exact value of $\sin (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$e^{\frac{3 π}{4}} \frac{\sqrt{2}}{2}$

Step 4.1.2.3

Combine $e^{\frac{3 π}{4}}$ and $\frac{\sqrt{2}}{2}$ .

$\frac{e^{\frac{3 π}{4}} \sqrt{2}}{2}$

Step 4.2

Evaluate at $x = \frac{7 π}{4}$ .

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Step 4.2.1

Substitute $\frac{7 π}{4}$ for $x$ .

$e^{\frac{7 π}{4}} \sin (\frac{7 π}{4})$

Step 4.2.2

Simplify.

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Step 4.2.2.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because sine is negative in the fourth quadrant.

$e^{\frac{7 π}{4}} (- \sin (\frac{π}{4}))$

Step 4.2.2.2

The exact value of $\sin (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$e^{\frac{7 π}{4}} (- \frac{\sqrt{2}}{2})$

Step 4.2.2.3

Combine $e^{\frac{7 π}{4}}$ and $\frac{\sqrt{2}}{2}$ .

$- \frac{e^{\frac{7 π}{4}} \sqrt{2}}{2}$

Step 4.3

Evaluate at $x = \frac{11 π}{4}$ .

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Step 4.3.1

Substitute $\frac{11 π}{4}$ for $x$ .

$e^{\frac{11 π}{4}} \sin (\frac{11 π}{4})$

Step 4.3.2

Simplify.

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Step 4.3.2.1

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$e^{\frac{11 π}{4}} \sin (\frac{3 π}{4})$

Step 4.3.2.2

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$e^{\frac{11 π}{4}} \sin (\frac{π}{4})$

Step 4.3.2.3

The exact value of $\sin (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$e^{\frac{11 π}{4}} \frac{\sqrt{2}}{2}$

Step 4.3.2.4

Combine $e^{\frac{11 π}{4}}$ and $\frac{\sqrt{2}}{2}$ .

$\frac{e^{\frac{11 π}{4}} \sqrt{2}}{2}$

Step 4.4

Evaluate at $x = \frac{15 π}{4}$ .

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Step 4.4.1

Substitute $\frac{15 π}{4}$ for $x$ .

$e^{\frac{15 π}{4}} \sin (\frac{15 π}{4})$

Step 4.4.2

Simplify.

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Step 4.4.2.1

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$e^{\frac{15 π}{4}} \sin (\frac{7 π}{4})$

Step 4.4.2.2

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because sine is negative in the fourth quadrant.

$e^{\frac{15 π}{4}} (- \sin (\frac{π}{4}))$

Step 4.4.2.3

The exact value of $\sin (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$e^{\frac{15 π}{4}} (- \frac{\sqrt{2}}{2})$

Step 4.4.2.4

Combine $e^{\frac{15 π}{4}}$ and $\frac{\sqrt{2}}{2}$ .

$- \frac{e^{\frac{15 π}{4}} \sqrt{2}}{2}$

Step 4.5

Evaluate at $x = \frac{19 π}{4}$ .

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Step 4.5.1

Substitute $\frac{19 π}{4}$ for $x$ .

$e^{\frac{19 π}{4}} \sin (\frac{19 π}{4})$

Step 4.5.2

Simplify.

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Step 4.5.2.1

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$e^{\frac{19 π}{4}} \sin (\frac{3 π}{4})$

Step 4.5.2.2

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$e^{\frac{19 π}{4}} \sin (\frac{π}{4})$

Step 4.5.2.3

The exact value of $\sin (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$e^{\frac{19 π}{4}} \frac{\sqrt{2}}{2}$

Step 4.5.2.4

Combine $e^{\frac{19 π}{4}}$ and $\frac{\sqrt{2}}{2}$ .

$\frac{e^{\frac{19 π}{4}} \sqrt{2}}{2}$

Step 4.6

List all of the points.

$(\frac{3 π}{4}, \frac{e^{\frac{3 π}{4}} \sqrt{2}}{2}), (\frac{7 π}{4}, - \frac{e^{\frac{7 π}{4}} \sqrt{2}}{2}), (\frac{11 π}{4}, \frac{e^{\frac{11 π}{4}} \sqrt{2}}{2}), (\frac{15 π}{4}, - \frac{e^{\frac{15 π}{4}} \sqrt{2}}{2}), (\frac{19 π}{4}, \frac{e^{\frac{19 π}{4}} \sqrt{2}}{2})$

Step 5