Calculus Examples

$y = x + \sin (x y)$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (x + \sin (x y))$

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.

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Step 3.1.1

By the Sum Rule, the derivative of $x + \sin (x y)$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [\sin (x y)]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [\sin (x y)]$

Step 3.1.2

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

$1 + \frac{d}{d x} [\sin (x y)]$

Step 3.2

Evaluate $\frac{d}{d x} [\sin (x y)]$ .

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Step 3.2.1

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

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Step 3.2.1.1

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

$1 + \frac{d}{d u} [\sin (u)] \frac{d}{d x} [x y]$

Step 3.2.1.2

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

$1 + \cos (u) \frac{d}{d x} [x y]$

Step 3.2.1.3

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

$1 + \cos (x y) \frac{d}{d x} [x y]$

Step 3.2.2

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

$1 + \cos (x y) (x \frac{d}{d x} [y] + y \frac{d}{d x} [x])$

Step 3.2.3

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

$1 + \cos (x y) (x y' + y \frac{d}{d x} [x])$

Step 3.2.4

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

$1 + \cos (x y) (x y' + y \cdot 1)$

Step 3.2.5

Multiply $y$ by $1$ .

$1 + \cos (x y) (x y' + y)$

Step 3.3

Simplify.

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Step 3.3.1

Apply the distributive property.

$1 + \cos (x y) (x y') + \cos (x y) y$

Step 3.3.2

Reorder terms.

$x \cos (x y) y' + y \cos (x y) + 1$

Step 4

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

$y' = x \cos (x y) y' + y \cos (x y) + 1$

Step 5

Solve for $y'$ .

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Step 5.1

Simplify the right side.

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Step 5.1.1

Reorder factors in $x \cos (x y) y' + y \cos (x y) + 1$ .

$y' = x y' \cos (x y) + y \cos (x y) + 1$

Step 5.2

Subtract $x y' \cos (x y)$ from both sides of the equation.

$y' - x y' \cos (x y) = y \cos (x y) + 1$

Step 5.3

Factor $y'$ out of $y' - x y' \cos (x y)$ .

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Step 5.3.1

Factor $y'$ out of ${y'}^{1}$ .

$y' \cdot 1 - x y' \cos (x y) = y \cos (x y) + 1$

Step 5.3.2

Factor $y'$ out of $- x y' \cos (x y)$ .

$y' \cdot 1 + y' (- x \cos (x y)) = y \cos (x y) + 1$

Step 5.3.3

Factor $y'$ out of $y' \cdot 1 + y' (- x \cos (x y))$ .

$y' (1 - x \cos (x y)) = y \cos (x y) + 1$

Step 5.4

Divide each term in $y' (1 - x \cos (x y)) = y \cos (x y) + 1$ by $1 - x \cos (x y)$ and simplify.

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Step 5.4.1

Divide each term in $y' (1 - x \cos (x y)) = y \cos (x y) + 1$ by $1 - x \cos (x y)$ .

$\frac{y' (1 - x \cos (x y))}{1 - x \cos (x y)} = \frac{y \cos (x y)}{1 - x \cos (x y)} + \frac{1}{1 - x \cos (x y)}$

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of $1 - x \cos (x y)$ .

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Step 5.4.2.1.1

Cancel the common factor.

$\frac{y' (1 - x \cos (x y))}{1 - x \cos (x y)} = \frac{y \cos (x y)}{1 - x \cos (x y)} + \frac{1}{1 - x \cos (x y)}$

Step 5.4.2.1.2

Divide $y'$ by $1$ .

$y' = \frac{y \cos (x y)}{1 - x \cos (x y)} + \frac{1}{1 - x \cos (x y)}$

Step 5.4.3

Simplify the right side.

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Step 5.4.3.1

Combine the numerators over the common denominator.

$y' = \frac{y \cos (x y) + 1}{1 - x \cos (x y)}$

Step 6

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

$\frac{d y}{d x} = \frac{y \cos (x y) + 1}{1 - x \cos (x y)}$

Step 7

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

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Step 7.1

Set the numerator equal to zero.

$y \cos (x y) + 1 = 0$

Step 7.2

Solve the equation for $x$ .

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Step 7.2.1

Subtract $1$ from both sides of the equation.

$y \cos (x y) = - 1$

Step 7.2.2

Divide each term in $y \cos (x y) = - 1$ by $y$ and simplify.

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Step 7.2.2.1

Divide each term in $y \cos (x y) = - 1$ by $y$ .

$\frac{y \cos (x y)}{y} = \frac{- 1}{y}$

Step 7.2.2.2

Simplify the left side.

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Step 7.2.2.2.1

Cancel the common factor of $y$ .

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Step 7.2.2.2.1.1

Cancel the common factor.

$\frac{y \cos (x y)}{y} = \frac{- 1}{y}$

Step 7.2.2.2.1.2

Divide $\cos (x y)$ by $1$ .

$\cos (x y) = \frac{- 1}{y}$

Step 7.2.2.3

Simplify the right side.

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Step 7.2.2.3.1

Move the negative in front of the fraction.

$\cos (x y) = - \frac{1}{y}$

Step 7.2.3

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

$x y = \arccos (- \frac{1}{y})$

Step 7.2.4

Divide each term in $x y = \arccos (- \frac{1}{y})$ by $y$ and simplify.

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Step 7.2.4.1

Divide each term in $x y = \arccos (- \frac{1}{y})$ by $y$ .

$\frac{x y}{y} = \frac{\arccos (- \frac{1}{y})}{y}$

Step 7.2.4.2

Simplify the left side.

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Step 7.2.4.2.1

Cancel the common factor of $y$ .

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Step 7.2.4.2.1.1

Cancel the common factor.

$\frac{x y}{y} = \frac{\arccos (- \frac{1}{y})}{y}$

Step 7.2.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{\arccos (- \frac{1}{y})}{y}$

Step 8

Solve for $y$ .

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Step 8.1

Simplify $(\frac{\arccos (- \frac{1}{y})}{y}) + \sin ((\frac{\arccos (- \frac{1}{y})}{y}) y)$ .

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Step 8.1.1

Simplify each term.

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Step 8.1.1.1

Cancel the common factor of $y$ .

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Step 8.1.1.1.1

Cancel the common factor.

$y = \frac{\arccos (- \frac{1}{y})}{y} + \sin (\frac{\arccos (- \frac{1}{y})}{y} y)$

Step 8.1.1.1.2

Rewrite the expression.

$y = \frac{\arccos (- \frac{1}{y})}{y} + \sin (\arccos (- \frac{1}{y}))$

Step 8.1.1.2

Draw a triangle in the plane with vertices $(- \frac{1}{y}, \sqrt{1^{2} - {(- \frac{1}{y})}^{2}})$ , $(- \frac{1}{y}, 0)$ , and the origin. Then $\arccos (- \frac{1}{y})$ is the angle between the positive x-axis and the ray beginning at the origin and passing through $(- \frac{1}{y}, \sqrt{1^{2} - {(- \frac{1}{y})}^{2}})$ . Therefore, $\sin (\arccos (- \frac{1}{y}))$ is $\sqrt{1 - {(- \frac{1}{y})}^{2}}$ .

$y = \frac{\arccos (- \frac{1}{y})}{y} + \sqrt{1 - {(- \frac{1}{y})}^{2}}$

Step 8.1.1.3

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

$y = \frac{\arccos (- \frac{1}{y})}{y} + \sqrt{1^{2} - {(- \frac{1}{y})}^{2}}$

Step 8.1.1.4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = - \frac{1}{y}$ .

$y = \frac{\arccos (- \frac{1}{y})}{y} + \sqrt{(1 - \frac{1}{y}) (1 - - \frac{1}{y})}$

Step 8.1.1.5

Multiply $- - \frac{1}{y}$ .

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Step 8.1.1.5.1

Multiply $- 1$ by $- 1$ .

$y = \frac{\arccos (- \frac{1}{y})}{y} + \sqrt{(1 - \frac{1}{y}) (1 + 1 \frac{1}{y})}$

Step 8.1.1.5.2

Multiply $\frac{1}{y}$ by $1$ .

$y = \frac{\arccos (- \frac{1}{y})}{y} + \sqrt{(1 - \frac{1}{y}) (1 + \frac{1}{y})}$

Step 8.1.1.6

Write $1$ as a fraction with a common denominator.

$y = \frac{\arccos (- \frac{1}{y})}{y} + \sqrt{(\frac{y}{y} - \frac{1}{y}) (1 + \frac{1}{y})}$

Step 8.1.1.7

Combine the numerators over the common denominator.

$y = \frac{\arccos (- \frac{1}{y})}{y} + \sqrt{\frac{y - 1}{y} (1 + \frac{1}{y})}$

Step 8.1.1.8

Write $1$ as a fraction with a common denominator.

$y = \frac{\arccos (- \frac{1}{y})}{y} + \sqrt{\frac{y - 1}{y} (\frac{y}{y} + \frac{1}{y})}$

Step 8.1.1.9

Combine the numerators over the common denominator.

$y = \frac{\arccos (- \frac{1}{y})}{y} + \sqrt{\frac{y - 1}{y} \cdot \frac{y + 1}{y}}$

Step 8.1.1.10

Multiply $\frac{y - 1}{y}$ by $\frac{y + 1}{y}$ .

$y = \frac{\arccos (- \frac{1}{y})}{y} + \sqrt{\frac{(y - 1) (y + 1)}{y \cdot y}}$

Step 8.1.1.11

Multiply $y$ by $y$ .

$y = \frac{\arccos (- \frac{1}{y})}{y} + \sqrt{\frac{(y - 1) (y + 1)}{y^{2}}}$

Step 8.1.1.12

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

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Step 8.1.1.12.1

Factor the perfect power $1^{2}$ out of $(y - 1) (y + 1)$ .

$y = \frac{\arccos (- \frac{1}{y})}{y} + \sqrt{\frac{1^{2} ((y - 1) (y + 1))}{y^{2}}}$

Step 8.1.1.12.2

Factor the perfect power $y^{2}$ out of $y^{2}$ .

$y = \frac{\arccos (- \frac{1}{y})}{y} + \sqrt{\frac{1^{2} ((y - 1) (y + 1))}{y^{2} \cdot 1}}$

Step 8.1.1.12.3

Rearrange the fraction $\frac{1^{2} ((y - 1) (y + 1))}{y^{2} \cdot 1}$ .

$y = \frac{\arccos (- \frac{1}{y})}{y} + \sqrt{{(\frac{1}{y})}^{2} ((y - 1) (y + 1))}$

Step 8.1.1.13

Pull terms out from under the radical.

$y = \frac{\arccos (- \frac{1}{y})}{y} + \frac{1}{y} \sqrt{(y - 1) (y + 1)}$

Step 8.1.1.14

Combine $\frac{1}{y}$ and $\sqrt{(y - 1) (y + 1)}$ .

$y = \frac{\arccos (- \frac{1}{y})}{y} + \frac{\sqrt{(y - 1) (y + 1)}}{y}$

Step 8.1.2

Combine the numerators over the common denominator.

$y = \frac{\arccos (- \frac{1}{y}) + \sqrt{(y - 1) (y + 1)}}{y}$

Step 8.2

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

$y \approx 1.94368519$

Step 9

Solve for $x$ when $y$ is $1.94368519$ .

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Step 9.1

Remove parentheses.

$x = \frac{\arccos (- \frac{1}{1.94368519})}{1.94368519}$

Step 9.2

Simplify $\frac{\arccos (- \frac{1}{1.94368519})}{1.94368519}$ .

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Step 9.2.1

Simplify the numerator.

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Step 9.2.1.1

Divide $1$ by $1.94368519$ .

$x = \frac{\arccos (- 1 \cdot 0.5144866)}{1.94368519}$

Step 9.2.1.2

Multiply $- 1$ by $0.5144866$ .

$x = \frac{\arccos (- 0.5144866)}{1.94368519}$

Step 9.2.1.3

Evaluate $\arccos (- 0.5144866)$ .

$x = \frac{2.11120515}{1.94368519}$

Step 9.2.2

Divide $2.11120515$ by $1.94368519$ .

$x = 1.08618677$

Step 10

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

$(1.08618677, 1.94368519)$

Step 11