Calculus Examples

$3 y^{2} - \sin (x y) - 5 \sqrt{x} = 0$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$3 y^{2} - \sin (x y) - 5 x^{\frac{1}{2}} = 0$

Step 2

Differentiate both sides of the equation.

$\frac{d}{d x} (3 y^{2} - \sin (x y) - 5 x^{\frac{1}{2}}) = \frac{d}{d x} (0)$

Step 3

Differentiate the left side of the equation.

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Step 3.1

By the Sum Rule, the derivative of $3 y^{2} - \sin (x y) - 5 x^{\frac{1}{2}}$ with respect to $x$ is $\frac{d}{d x} [3 y^{2}] + \frac{d}{d x} [- \sin (x y)] + \frac{d}{d x} [- 5 x^{\frac{1}{2}}]$ .

$\frac{d}{d x} [3 y^{2}] + \frac{d}{d x} [- \sin (x y)] + \frac{d}{d x} [- 5 x^{\frac{1}{2}}]$

Step 3.2

Evaluate $\frac{d}{d x} [3 y^{2}]$ .

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Step 3.2.1

Since $3$ is constant with respect to $x$ , the derivative of $3 y^{2}$ with respect to $x$ is $3 \frac{d}{d x} [y^{2}]$ .

$3 \frac{d}{d x} [y^{2}] + \frac{d}{d x} [- \sin (x y)] + \frac{d}{d x} [- 5 x^{\frac{1}{2}}]$

Step 3.2.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = y$ .

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Step 3.2.2.1

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

$3 (\frac{d}{d u_{1}} [{u_{1}}^{2}] \frac{d}{d x} [y]) + \frac{d}{d x} [- \sin (x y)] + \frac{d}{d x} [- 5 x^{\frac{1}{2}}]$

Step 3.2.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = 2$ .

$3 (2 u_{1} \frac{d}{d x} [y]) + \frac{d}{d x} [- \sin (x y)] + \frac{d}{d x} [- 5 x^{\frac{1}{2}}]$

Step 3.2.2.3

Replace all occurrences of $u_{1}$ with $y$ .

$3 (2 y \frac{d}{d x} [y]) + \frac{d}{d x} [- \sin (x y)] + \frac{d}{d x} [- 5 x^{\frac{1}{2}}]$

Step 3.2.3

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

$3 (2 y y') + \frac{d}{d x} [- \sin (x y)] + \frac{d}{d x} [- 5 x^{\frac{1}{2}}]$

Step 3.2.4

Multiply $2$ by $3$ .

$6 y y' + \frac{d}{d x} [- \sin (x y)] + \frac{d}{d x} [- 5 x^{\frac{1}{2}}]$

Step 3.3

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

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Step 3.3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- \sin (x y)$ with respect to $x$ is $- \frac{d}{d x} [\sin (x y)]$ .

$6 y y' - \frac{d}{d x} [\sin (x y)] + \frac{d}{d x} [- 5 x^{\frac{1}{2}}]$

Step 3.3.2

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

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Step 3.3.2.1

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

$6 y y' - (\frac{d}{d u_{2}} [\sin (u_{2})] \frac{d}{d x} [x y]) + \frac{d}{d x} [- 5 x^{\frac{1}{2}}]$

Step 3.3.2.2

The derivative of $\sin (u_{2})$ with respect to $u_{2}$ is $\cos (u_{2})$ .

$6 y y' - (\cos (u_{2}) \frac{d}{d x} [x y]) + \frac{d}{d x} [- 5 x^{\frac{1}{2}}]$

Step 3.3.2.3

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

$6 y y' - (\cos (x y) \frac{d}{d x} [x y]) + \frac{d}{d x} [- 5 x^{\frac{1}{2}}]$

Step 3.3.3

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 $g (x) = y$ .

$6 y y' - (\cos (x y) (x \frac{d}{d x} [y] + y \frac{d}{d x} [x])) + \frac{d}{d x} [- 5 x^{\frac{1}{2}}]$

Step 3.3.4

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

$6 y y' - (\cos (x y) (x y' + y \frac{d}{d x} [x])) + \frac{d}{d x} [- 5 x^{\frac{1}{2}}]$

Step 3.3.5

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

$6 y y' - (\cos (x y) (x y' + y \cdot 1)) + \frac{d}{d x} [- 5 x^{\frac{1}{2}}]$

Step 3.3.6

Multiply $y$ by $1$ .

$6 y y' - \cos (x y) (x y' + y) + \frac{d}{d x} [- 5 x^{\frac{1}{2}}]$

Step 3.4

Evaluate $\frac{d}{d x} [- 5 x^{\frac{1}{2}}]$ .

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Step 3.4.1

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5 x^{\frac{1}{2}}$ with respect to $x$ is $- 5 \frac{d}{d x} [x^{\frac{1}{2}}]$ .

$6 y y' - \cos (x y) (x y' + y) - 5 \frac{d}{d x} [x^{\frac{1}{2}}]$

Step 3.4.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{1}{2}$ .

$6 y y' - \cos (x y) (x y' + y) - 5 (\frac{1}{2} x^{\frac{1}{2} - 1})$

Step 3.4.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$6 y y' - \cos (x y) (x y' + y) - 5 (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}})$

Step 3.4.4

Combine $- 1$ and $\frac{2}{2}$ .

$6 y y' - \cos (x y) (x y' + y) - 5 (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}})$

Step 3.4.5

Combine the numerators over the common denominator.

$6 y y' - \cos (x y) (x y' + y) - 5 (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}})$

Step 3.4.6

Simplify the numerator.

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Step 3.4.6.1

Multiply $- 1$ by $2$ .

$6 y y' - \cos (x y) (x y' + y) - 5 (\frac{1}{2} x^{\frac{1 - 2}{2}})$

Step 3.4.6.2

Subtract $2$ from $1$ .

$6 y y' - \cos (x y) (x y' + y) - 5 (\frac{1}{2} x^{\frac{- 1}{2}})$

Step 3.4.7

Move the negative in front of the fraction.

$6 y y' - \cos (x y) (x y' + y) - 5 (\frac{1}{2} x^{- \frac{1}{2}})$

Step 3.4.8

Combine $\frac{1}{2}$ and $x^{- \frac{1}{2}}$ .

$6 y y' - \cos (x y) (x y' + y) - 5 \frac{x^{- \frac{1}{2}}}{2}$

Step 3.4.9

Combine $- 5$ and $\frac{x^{- \frac{1}{2}}}{2}$ .

$6 y y' - \cos (x y) (x y' + y) + \frac{- 5 x^{- \frac{1}{2}}}{2}$

Step 3.4.10

Move $x^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$6 y y' - \cos (x y) (x y' + y) + \frac{- 5}{2 x^{\frac{1}{2}}}$

Step 3.4.11

Move the negative in front of the fraction.

$6 y y' - \cos (x y) (x y' + y) - \frac{5}{2 x^{\frac{1}{2}}}$

Step 3.5

Simplify.

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Step 3.5.1

Apply the distributive property.

$6 y y' - \cos (x y) (x y') - \cos (x y) y - \frac{5}{2 x^{\frac{1}{2}}}$

Step 3.5.2

Reorder terms.

$6 y y' - x \cos (x y) y' - y \cos (x y) - \frac{5}{2 x^{\frac{1}{2}}}$

Step 4

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

$0$

Step 5

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

$6 y y' - x \cos (x y) y' - y \cos (x y) - \frac{5}{2 x^{\frac{1}{2}}} = 0$

Step 6

Solve for $y'$ .

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Step 6.1

Simplify the left side.

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Step 6.1.1

Reorder factors in $6 y y' - x \cos (x y) y' - y \cos (x y) - \frac{5}{2 x^{\frac{1}{2}}}$ .

$6 y y' - x y' \cos (x y) - y \cos (x y) - \frac{5}{2 x^{\frac{1}{2}}} = 0$

Step 6.2

Move all terms not containing $y'$ to the right side of the equation.

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Step 6.2.1

Add $y \cos (x y)$ to both sides of the equation.

$6 y y' - x y' \cos (x y) - \frac{5}{2 x^{\frac{1}{2}}} = y \cos (x y)$

Step 6.2.2

Add $\frac{5}{2 x^{\frac{1}{2}}}$ to both sides of the equation.

$6 y y' - x y' \cos (x y) = y \cos (x y) + \frac{5}{2 x^{\frac{1}{2}}}$

Step 6.3

Factor $y'$ out of $6 y y' - x y' \cos (x y)$ .

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Step 6.3.1

Factor $y'$ out of $6 y y'$ .

$y' (6 y) - x y' \cos (x y) = y \cos (x y) + \frac{5}{2 x^{\frac{1}{2}}}$

Step 6.3.2

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

$y' (6 y) + y' (- x \cos (x y)) = y \cos (x y) + \frac{5}{2 x^{\frac{1}{2}}}$

Step 6.3.3

Factor $y'$ out of $y' (6 y) + y' (- x \cos (x y))$ .

$y' (6 y - x \cos (x y)) = y \cos (x y) + \frac{5}{2 x^{\frac{1}{2}}}$

Step 6.4

Divide each term in $y' (6 y - x \cos (x y)) = y \cos (x y) + \frac{5}{2 x^{\frac{1}{2}}}$ by $6 y - x \cos (x y)$ and simplify.

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Step 6.4.1

Divide each term in $y' (6 y - x \cos (x y)) = y \cos (x y) + \frac{5}{2 x^{\frac{1}{2}}}$ by $6 y - x \cos (x y)$ .

$\frac{y' (6 y - x \cos (x y))}{6 y - x \cos (x y)} = \frac{y \cos (x y)}{6 y - x \cos (x y)} + \frac{\frac{5}{2 x^{\frac{1}{2}}}}{6 y - x \cos (x y)}$

Step 6.4.2

Simplify the left side.

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Step 6.4.2.1

Cancel the common factor of $6 y - x \cos (x y)$ .

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Step 6.4.2.1.1

Cancel the common factor.

$\frac{y' (6 y - x \cos (x y))}{6 y - x \cos (x y)} = \frac{y \cos (x y)}{6 y - x \cos (x y)} + \frac{\frac{5}{2 x^{\frac{1}{2}}}}{6 y - x \cos (x y)}$

Step 6.4.2.1.2

Divide $y'$ by $1$ .

$y' = \frac{y \cos (x y)}{6 y - x \cos (x y)} + \frac{\frac{5}{2 x^{\frac{1}{2}}}}{6 y - x \cos (x y)}$

Step 6.4.3

Simplify the right side.

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Step 6.4.3.1

Simplify each term.

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Step 6.4.3.1.1

Multiply the numerator by the reciprocal of the denominator.

$y' = \frac{y \cos (x y)}{6 y - x \cos (x y)} + \frac{5}{2 x^{\frac{1}{2}}} \cdot \frac{1}{6 y - x \cos (x y)}$

Step 6.4.3.1.2

Multiply $\frac{5}{2 x^{\frac{1}{2}}}$ by $\frac{1}{6 y - x \cos (x y)}$ .

$y' = \frac{y \cos (x y)}{6 y - x \cos (x y)} + \frac{5}{2 x^{\frac{1}{2}} (6 y - x \cos (x y))}$

Step 6.4.3.2

To write $\frac{y \cos (x y)}{6 y - x \cos (x y)}$ as a fraction with a common denominator, multiply by $\frac{2 x^{\frac{1}{2}}}{2 x^{\frac{1}{2}}}$ .

$y' = \frac{y \cos (x y)}{6 y - x \cos (x y)} \cdot \frac{2 x^{\frac{1}{2}}}{2 x^{\frac{1}{2}}} + \frac{5}{2 x^{\frac{1}{2}} (6 y - x \cos (x y))}$

Step 6.4.3.3

Write each expression with a common denominator of $(6 y - x \cos (x y)) \cdot 2 x^{\frac{1}{2}}$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.4.3.3.1

Multiply $\frac{y \cos (x y)}{6 y - x \cos (x y)}$ by $\frac{2 x^{\frac{1}{2}}}{2 x^{\frac{1}{2}}}$ .

$y' = \frac{y \cos (x y) (2 x^{\frac{1}{2}})}{(6 y - x \cos (x y)) (2 x^{\frac{1}{2}})} + \frac{5}{2 x^{\frac{1}{2}} (6 y - x \cos (x y))}$

Step 6.4.3.3.2

Reorder the factors of $(6 y - x \cos (x y)) (2 x^{\frac{1}{2}})$ .

$y' = \frac{y \cos (x y) (2 x^{\frac{1}{2}})}{2 x^{\frac{1}{2}} (6 y - x \cos (x y))} + \frac{5}{2 x^{\frac{1}{2}} (6 y - x \cos (x y))}$

Step 6.4.3.4

Combine the numerators over the common denominator.

$y' = \frac{y \cos (x y) (2 x^{\frac{1}{2}}) + 5}{2 x^{\frac{1}{2}} (6 y - x \cos (x y))}$

Step 6.4.3.5

Simplify the expression.

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Step 6.4.3.5.1

Rewrite using the commutative property of multiplication.

$y' = \frac{2 y \cos (x y) x^{\frac{1}{2}} + 5}{2 x^{\frac{1}{2}} (6 y - x \cos (x y))}$

Step 6.4.3.5.2

Reorder factors in $\frac{2 y \cos (x y) x^{\frac{1}{2}} + 5}{2 x^{\frac{1}{2}} (6 y - x \cos (x y))}$ .

$y' = \frac{2 y x^{\frac{1}{2}} \cos (x y) + 5}{2 x^{\frac{1}{2}} (6 y - x \cos (x y))}$

Step 7

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

$\frac{d y}{d x} = \frac{2 y x^{\frac{1}{2}} \cos (x y) + 5}{2 x^{\frac{1}{2}} (6 y - x \cos (x y))}$