Calculus Examples

$x = \tan (y)$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (x) = \frac{d}{d x} (\tan (y))$

Step 2

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

$1$

Step 3

Differentiate the right side of the equation.

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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 $f (x) = \tan (x)$ and $g (x) = y$ .

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Step 3.1.1

To apply the Chain Rule, set $u$ as $y$ .

$\frac{d}{d u} [\tan (u)] \frac{d}{d x} [y]$

Step 3.1.2

The derivative of $\tan (u)$ with respect to $u$ is $\sec^{2} (u)$ .

$\sec^{2} (u) \frac{d}{d x} [y]$

Step 3.1.3

Replace all occurrences of $u$ with $y$ .

$\sec^{2} (y) \frac{d}{d x} [y]$

Step 3.2

Rewrite $\frac{d}{d x} [y]$ as $y'$ .

$\sec^{2} (y) y'$

Step 4

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

$1 = \sec^{2} (y) y'$

Step 5

Solve for $y'$ .

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Step 5.1

Rewrite the equation as $\sec^{2} (y) y' = 1$ .

$\sec^{2} (y) y' = 1$

Step 5.2

Divide each term in $\sec^{2} (y) y' = 1$ by $\sec^{2} (y)$ and simplify.

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Step 5.2.1

Divide each term in $\sec^{2} (y) y' = 1$ by $\sec^{2} (y)$ .

$\frac{\sec^{2} (y) y'}{\sec^{2} (y)} = \frac{1}{\sec^{2} (y)}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $\sec^{2} (y)$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{\sec^{2} (y) y'}{\sec^{2} (y)} = \frac{1}{\sec^{2} (y)}$

Step 5.2.2.1.2

Divide $y'$ by $1$ .

$y' = \frac{1}{\sec^{2} (y)}$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Rewrite $1$ as $1^{2}$ .

$y' = \frac{1^{2}}{\sec^{2} (y)}$

Step 5.2.3.2

Rewrite $\frac{1^{2}}{\sec^{2} (y)}$ as ${(\frac{1}{\sec (y)})}^{2}$ .

$y' = {(\frac{1}{\sec (y)})}^{2}$

Step 5.2.3.3

Rewrite $\sec (y)$ in terms of sines and cosines.

$y' = {(\frac{1}{\frac{1}{\cos (y)}})}^{2}$

Step 5.2.3.4

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\cos (y)}$ .

$y' = {(1 \cos (y))}^{2}$

Step 5.2.3.5

Multiply $\cos (y)$ by $1$ .

$y' = \cos^{2} (y)$

Step 6

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

$\frac{d y}{d x} = \cos^{2} (y)$

Step 7

Set $\frac{d y}{d x} = 0$ then solve for $x$ in terms of $y$ .

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Step 7.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$\cos (y) = \pm \sqrt{0}$

Step 7.2

Simplify $\pm \sqrt{0}$ .

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Step 7.2.1

Rewrite $0$ as $0^{2}$ .

$\cos (y) = \pm \sqrt{0^{2}}$

Step 7.2.2

Pull terms out from under the radical, assuming positive real numbers.

$\cos (y) = \pm 0$

Step 7.2.3

Plus or minus $0$ is $0$ .

$\cos (y) = 0$

Step 7.3

Take the inverse cosine of both sides of the equation to extract $y$ from inside the cosine.

$y = \arccos (0)$

Step 7.4

Simplify the right side.

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Step 7.4.1

The exact value of $\arccos (0)$ is $\frac{π}{2}$ .

$y = \frac{π}{2}$

Step 7.5

The cosine function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the fourth quadrant.

$y = 2 π - \frac{π}{2}$

Step 7.6

Simplify $2 π - \frac{π}{2}$ .

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Step 7.6.1

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

$y = 2 π \cdot \frac{2}{2} - \frac{π}{2}$

Step 7.6.2

Combine fractions.

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Step 7.6.2.1

Combine $2 π$ and $\frac{2}{2}$ .

$y = \frac{2 π \cdot 2}{2} - \frac{π}{2}$

Step 7.6.2.2

Combine the numerators over the common denominator.

$y = \frac{2 π \cdot 2 - π}{2}$

Step 7.6.3

Simplify the numerator.

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Step 7.6.3.1

Multiply $2$ by $2$ .

$y = \frac{4 π - π}{2}$

Step 7.6.3.2

Subtract $π$ from $4 π$ .

$y = \frac{3 π}{2}$

Step 7.7

Find the period of $\cos (y)$ .

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Step 7.7.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 7.7.2

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

$\frac{2 π}{| 1 |}$

Step 7.7.3

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

$\frac{2 π}{1}$

Step 7.7.4

Divide $2 π$ by $1$ .

$2 π$

Step 7.8

The period of the $\cos (y)$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

$y = \frac{π}{2} + 2 π n, \frac{3 π}{2} + 2 π n$ , for any integer $n$

Step 7.9

Consolidate the answers.

$y = \frac{π}{2} + π n$ , for any integer $n$

Step 8

Solve for $y$ .

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Step 8.1

Rewrite the equation as $\tan (y) = \frac{π}{2} + π n$ .

$\tan (y) = \frac{π}{2} + π n$

Step 8.2

Reorder $\frac{π}{2}$ and $π n$ .

$\tan (y) = π n + \frac{π}{2}$

Step 8.3

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

$y = \arctan (π n + \frac{π}{2})$

Step 9

Solve for $x$ when $y$ is $\arctan (π n + \frac{π}{2})$ .

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Step 9.1

Remove parentheses.

$\arctan (π n + \frac{π}{2}) = \frac{π}{2} + π n$

Step 9.2

Graph each side of the equation. The solution is the x-value of the point of intersection.

$n = - \frac{1}{2}$

Step 10

Find the points where $\frac{d y}{d x} = 0$ .

$(- \frac{1}{2}, \arctan (π n + \frac{π}{2}))$

Step 11