Calculus Examples

$x + 2 y = \sqrt{y}$

Step 1

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

$x + 2 y = y^{\frac{1}{2}}$

Step 2

Differentiate both sides of the equation.

$\frac{d}{d x} (x + 2 y) = \frac{d}{d x} (y^{\frac{1}{2}})$

Step 3

Differentiate the left 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 $x + 2 y$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [2 y]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [2 y]$

Step 3.1.2

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

$1 + \frac{d}{d x} [2 y]$

Step 3.2

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

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Step 3.2.1

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

$1 + 2 \frac{d}{d x} [y]$

Step 3.2.2

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

$1 + 2 y'$

Step 3.3

Reorder terms.

$2 y' + 1$

Step 4

Differentiate the right side of the equation.

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Step 4.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = y$ .

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Step 4.1.1

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

$\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [y]$

Step 4.1.2

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

$\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [y]$

Step 4.1.3

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

$\frac{1}{2} y^{\frac{1}{2} - 1} \frac{d}{d x} [y]$

Step 4.2

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

$\frac{1}{2} y^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [y]$

Step 4.3

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

$\frac{1}{2} y^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [y]$

Step 4.4

Combine the numerators over the common denominator.

$\frac{1}{2} y^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [y]$

Step 4.5

Simplify the numerator.

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Step 4.5.1

Multiply $- 1$ by $2$ .

$\frac{1}{2} y^{\frac{1 - 2}{2}} \frac{d}{d x} [y]$

Step 4.5.2

Subtract $2$ from $1$ .

$\frac{1}{2} y^{\frac{- 1}{2}} \frac{d}{d x} [y]$

Step 4.6

Move the negative in front of the fraction.

$\frac{1}{2} y^{- \frac{1}{2}} \frac{d}{d x} [y]$

Step 4.7

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

$\frac{y^{- \frac{1}{2}}}{2} \frac{d}{d x} [y]$

Step 4.8

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

$\frac{1}{2 y^{\frac{1}{2}}} \frac{d}{d x} [y]$

Step 4.9

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

$\frac{1}{2 y^{\frac{1}{2}}} y'$

Step 4.10

Combine $\frac{1}{2 y^{\frac{1}{2}}}$ and $y'$ .

$\frac{y'}{2 y^{\frac{1}{2}}}$

Step 5

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

$2 y' + 1 = \frac{y'}{2 y^{\frac{1}{2}}}$

Step 6

Solve for $y'$ .

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Step 6.1

Multiply both sides by $2 y^{\frac{1}{2}}$ .

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

Step 6.2

Simplify.

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Step 6.2.1

Simplify the left side.

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Step 6.2.1.1

Simplify $(2 y' + 1) (2 y^{\frac{1}{2}})$ .

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Step 6.2.1.1.1

Apply the distributive property.

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

Step 6.2.1.1.2

Simplify the expression.

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Step 6.2.1.1.2.1

Rewrite using the commutative property of multiplication.

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

Step 6.2.1.1.2.2

Multiply $2 y^{\frac{1}{2}}$ by $1$ .

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

Step 6.2.1.1.2.3

Multiply $2$ by $2$ .

$4 y' y^{\frac{1}{2}} + 2 y^{\frac{1}{2}} = \frac{y'}{2 y^{\frac{1}{2}}} (2 y^{\frac{1}{2}})$

Step 6.2.2

Simplify the right side.

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Step 6.2.2.1

Simplify $\frac{y'}{2 y^{\frac{1}{2}}} (2 y^{\frac{1}{2}})$ .

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Step 6.2.2.1.1

Rewrite using the commutative property of multiplication.

$4 y' y^{\frac{1}{2}} + 2 y^{\frac{1}{2}} = 2 \frac{y'}{2 y^{\frac{1}{2}}} y^{\frac{1}{2}}$

Step 6.2.2.1.2

Cancel the common factor of $2$ .

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Step 6.2.2.1.2.1

Cancel the common factor.

$4 y' y^{\frac{1}{2}} + 2 y^{\frac{1}{2}} = 2 \frac{y'}{2 y^{\frac{1}{2}}} y^{\frac{1}{2}}$

Step 6.2.2.1.2.2

Rewrite the expression.

$4 y' y^{\frac{1}{2}} + 2 y^{\frac{1}{2}} = \frac{y'}{y^{\frac{1}{2}}} y^{\frac{1}{2}}$

Step 6.2.2.1.3

Cancel the common factor of $y^{\frac{1}{2}}$ .

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Step 6.2.2.1.3.1

Cancel the common factor.

$4 y' y^{\frac{1}{2}} + 2 y^{\frac{1}{2}} = \frac{y'}{y^{\frac{1}{2}}} y^{\frac{1}{2}}$

Step 6.2.2.1.3.2

Rewrite the expression.

$4 y' y^{\frac{1}{2}} + 2 y^{\frac{1}{2}} = y'$

Step 6.3

Solve for $y'$ .

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Step 6.3.1

Subtract $2 y^{\frac{1}{2}}$ from both sides of the equation.

$4 y' y^{\frac{1}{2}} - y' = - 2 y^{\frac{1}{2}}$

Step 6.3.2

Factor $y'$ out of $4 y' y^{\frac{1}{2}} - y'$ .

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Step 6.3.2.1

Factor $y'$ out of $4 y' y^{\frac{1}{2}}$ .

$y' (4 y^{\frac{1}{2}}) - y' = - 2 y^{\frac{1}{2}}$

Step 6.3.2.2

Factor $y'$ out of $- y'$ .

$y' (4 y^{\frac{1}{2}}) + y' \cdot - 1 = - 2 y^{\frac{1}{2}}$

Step 6.3.2.3

Factor $y'$ out of $y' (4 y^{\frac{1}{2}}) + y' \cdot - 1$ .

$y' (4 y^{\frac{1}{2}} - 1) = - 2 y^{\frac{1}{2}}$

Step 6.3.3

Divide each term in $y' (4 y^{\frac{1}{2}} - 1) = - 2 y^{\frac{1}{2}}$ by $4 y^{\frac{1}{2}} - 1$ and simplify.

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Step 6.3.3.1

Divide each term in $y' (4 y^{\frac{1}{2}} - 1) = - 2 y^{\frac{1}{2}}$ by $4 y^{\frac{1}{2}} - 1$ .

$\frac{y' (4 y^{\frac{1}{2}} - 1)}{4 y^{\frac{1}{2}} - 1} = \frac{- 2 y^{\frac{1}{2}}}{4 y^{\frac{1}{2}} - 1}$

Step 6.3.3.2

Simplify the left side.

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Step 6.3.3.2.1

Cancel the common factor.

$\frac{y' (4 y^{\frac{1}{2}} - 1)}{4 y^{\frac{1}{2}} - 1} = \frac{- 2 y^{\frac{1}{2}}}{4 y^{\frac{1}{2}} - 1}$

Step 6.3.3.2.2

Divide $y'$ by $1$ .

$y' = \frac{- 2 y^{\frac{1}{2}}}{4 y^{\frac{1}{2}} - 1}$

Step 6.3.3.3

Simplify the right side.

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Step 6.3.3.3.1

Move the negative in front of the fraction.

$y' = - \frac{2 y^{\frac{1}{2}}}{4 y^{\frac{1}{2}} - 1}$

Step 7

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

$\frac{d y}{d x} = - \frac{2 y^{\frac{1}{2}}}{4 y^{\frac{1}{2}} - 1}$