Calculus Examples

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

Step 1

Rewrite the left side with rational exponents.

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Step 1.1

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

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

Step 1.2

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

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

Step 2

Differentiate both sides of the equation.

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

Step 3

Differentiate the left side of the equation.

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Step 3.1

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

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

Step 3.2

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

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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 $f (x) = x^{\frac{1}{2}}$ and $g (x) = y$ .

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Step 3.2.1.1

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

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

Step 3.2.1.2

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

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

Step 3.2.1.3

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

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

Step 3.2.2

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

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

Step 3.2.3

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

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

Step 3.2.4

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

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

Step 3.2.5

Combine the numerators over the common denominator.

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

Step 3.2.6

Simplify the numerator.

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Step 3.2.6.1

Multiply $- 1$ by $2$ .

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

Step 3.2.6.2

Subtract $2$ from $1$ .

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

Step 3.2.7

Move the negative in front of the fraction.

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

Step 3.2.8

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

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

Step 3.2.9

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

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

Step 3.2.10

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

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

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$\frac{y'}{2 y^{\frac{1}{2}}} + \frac{y^{\frac{1}{2}} \cdot 0 - 1 \cdot 1 \frac{d}{d x} [y^{\frac{1}{2}}]}{{(y^{\frac{1}{2}})}^{2}}$

Step 3.3.3

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 3.3.3.1

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

$\frac{y'}{2 y^{\frac{1}{2}}} + \frac{y^{\frac{1}{2}} \cdot 0 - 1 \cdot 1 (\frac{d}{d u_{2}} [{u_{2}}^{\frac{1}{2}}] \frac{d}{d x} [y])}{{(y^{\frac{1}{2}})}^{2}}$

Step 3.3.3.2

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

$\frac{y'}{2 y^{\frac{1}{2}}} + \frac{y^{\frac{1}{2}} \cdot 0 - 1 \cdot 1 (\frac{1}{2} {u_{2}}^{\frac{1}{2} - 1} \frac{d}{d x} [y])}{{(y^{\frac{1}{2}})}^{2}}$

Step 3.3.3.3

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

$\frac{y'}{2 y^{\frac{1}{2}}} + \frac{y^{\frac{1}{2}} \cdot 0 - 1 \cdot 1 (\frac{1}{2} y^{\frac{1}{2} - 1} \frac{d}{d x} [y])}{{(y^{\frac{1}{2}})}^{2}}$

Step 3.3.4

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

$\frac{y'}{2 y^{\frac{1}{2}}} + \frac{y^{\frac{1}{2}} \cdot 0 - 1 \cdot 1 (\frac{1}{2} y^{\frac{1}{2} - 1} y')}{{(y^{\frac{1}{2}})}^{2}}$

Step 3.3.5

Multiply $y^{\frac{1}{2}}$ by $0$ .

$\frac{y'}{2 y^{\frac{1}{2}}} + \frac{0 - 1 \cdot 1 (\frac{1}{2} y^{\frac{1}{2} - 1} y')}{{(y^{\frac{1}{2}})}^{2}}$

Step 3.3.6

Multiply $- 1$ by $1$ .

$\frac{y'}{2 y^{\frac{1}{2}}} + \frac{0 - (\frac{1}{2} y^{\frac{1}{2} - 1} y')}{{(y^{\frac{1}{2}})}^{2}}$

Step 3.3.7

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

$\frac{y'}{2 y^{\frac{1}{2}}} + \frac{0 - (\frac{1}{2} y^{\frac{1}{2} - 1 \cdot \frac{2}{2}} y')}{{(y^{\frac{1}{2}})}^{2}}$

Step 3.3.8

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

$\frac{y'}{2 y^{\frac{1}{2}}} + \frac{0 - (\frac{1}{2} y^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} y')}{{(y^{\frac{1}{2}})}^{2}}$

Step 3.3.9

Combine the numerators over the common denominator.

$\frac{y'}{2 y^{\frac{1}{2}}} + \frac{0 - (\frac{1}{2} y^{\frac{1 - 1 \cdot 2}{2}} y')}{{(y^{\frac{1}{2}})}^{2}}$

Step 3.3.10

Simplify the numerator.

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Step 3.3.10.1

Multiply $- 1$ by $2$ .

$\frac{y'}{2 y^{\frac{1}{2}}} + \frac{0 - (\frac{1}{2} y^{\frac{1 - 2}{2}} y')}{{(y^{\frac{1}{2}})}^{2}}$

Step 3.3.10.2

Subtract $2$ from $1$ .

$\frac{y'}{2 y^{\frac{1}{2}}} + \frac{0 - (\frac{1}{2} y^{\frac{- 1}{2}} y')}{{(y^{\frac{1}{2}})}^{2}}$

Step 3.3.11

Move the negative in front of the fraction.

$\frac{y'}{2 y^{\frac{1}{2}}} + \frac{0 - (\frac{1}{2} y^{- \frac{1}{2}} y')}{{(y^{\frac{1}{2}})}^{2}}$

Step 3.3.12

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

$\frac{y'}{2 y^{\frac{1}{2}}} + \frac{0 - (\frac{y^{- \frac{1}{2}}}{2} y')}{{(y^{\frac{1}{2}})}^{2}}$

Step 3.3.13

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

$\frac{y'}{2 y^{\frac{1}{2}}} + \frac{0 - \frac{y^{- \frac{1}{2}} y'}{2}}{{(y^{\frac{1}{2}})}^{2}}$

Step 3.3.14

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

$\frac{y'}{2 y^{\frac{1}{2}}} + \frac{0 - \frac{y'}{2 y^{\frac{1}{2}}}}{{(y^{\frac{1}{2}})}^{2}}$

Step 3.3.15

Subtract $\frac{y'}{2 y^{\frac{1}{2}}}$ from $0$ .

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

Step 3.3.16

Multiply the exponents in ${(y^{\frac{1}{2}})}^{2}$ .

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Step 3.3.16.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{y'}{2 y^{\frac{1}{2}}} + \frac{- \frac{y'}{2 y^{\frac{1}{2}}}}{y^{\frac{1}{2} \cdot 2}}$

Step 3.3.16.2

Cancel the common factor of $2$ .

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Step 3.3.16.2.1

Cancel the common factor.

$\frac{y'}{2 y^{\frac{1}{2}}} + \frac{- \frac{y'}{2 y^{\frac{1}{2}}}}{y^{\frac{1}{2} \cdot 2}}$

Step 3.3.16.2.2

Rewrite the expression.

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

Step 3.3.17

Simplify.

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

Step 3.3.18

Rewrite $\frac{- \frac{y'}{2 y^{\frac{1}{2}}}}{y}$ as a product.

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

Step 3.3.19

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

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

Step 3.3.20

Raise $y$ to the power of $1$ .

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

Step 3.3.21

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 3.3.22

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

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

Step 3.3.23

Combine the numerators over the common denominator.

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

Step 3.3.24

Add $2$ and $1$ .

$\frac{y'}{2 y^{\frac{1}{2}}} - \frac{y'}{2 y^{\frac{3}{2}}}$

Step 4

Differentiate the right side of the equation.

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Step 4.1

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

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

Step 4.2

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

$2 \cdot 1$

Step 4.3

Multiply $2$ by $1$ .

$2$

Step 5

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

$\frac{y'}{2 y^{\frac{1}{2}}} - \frac{y'}{2 y^{\frac{3}{2}}} = 2$

Step 6

Solve for $y'$ .

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Step 6.1

Find the LCD of the terms in the equation.

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Step 6.1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$2 y^{\frac{1}{2}}, 2 y^{\frac{3}{2}}, 1$

Step 6.1.2

Since $2 y^{\frac{1}{2}}, 2 y^{\frac{3}{2}}, 1$ contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part $2, 2, 1$ then find LCM for the variable part $y^{\frac{1}{2}}, y^{\frac{3}{2}}$ .

Step 6.1.3

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

Step 6.1.4

Since $2$ has no factors besides $1$ and $2$ .

$2$ is a prime number

Step 6.1.5

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

Step 6.1.6

The LCM of $2, 2, 1$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$2$

Step 6.1.7

The LCM of $y^{\frac{1}{2}}, y^{\frac{3}{2}}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

$y^{\frac{3}{2}}$

Step 6.1.8

The LCM for $2 y^{\frac{1}{2}}, 2 y^{\frac{3}{2}}, 1$ is the numeric part $2$ multiplied by the variable part.

$2 y^{\frac{3}{2}}$

Step 6.2

Multiply each term in $\frac{y'}{2 y^{\frac{1}{2}}} - \frac{y'}{2 y^{\frac{3}{2}}} = 2$ by $2 y^{\frac{3}{2}}$ to eliminate the fractions.

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Step 6.2.1

Multiply each term in $\frac{y'}{2 y^{\frac{1}{2}}} - \frac{y'}{2 y^{\frac{3}{2}}} = 2$ by $2 y^{\frac{3}{2}}$ .

$\frac{y'}{2 y^{\frac{1}{2}}} (2 y^{\frac{3}{2}}) - \frac{y'}{2 y^{\frac{3}{2}}} (2 y^{\frac{3}{2}}) = 2 (2 y^{\frac{3}{2}})$

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

Simplify each term.

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Step 6.2.2.1.1

Rewrite using the commutative property of multiplication.

$2 \frac{y'}{2 y^{\frac{1}{2}}} y^{\frac{3}{2}} - \frac{y'}{2 y^{\frac{3}{2}}} (2 y^{\frac{3}{2}}) = 2 (2 y^{\frac{3}{2}})$

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.

$2 \frac{y'}{2 y^{\frac{1}{2}}} y^{\frac{3}{2}} - \frac{y'}{2 y^{\frac{3}{2}}} (2 y^{\frac{3}{2}}) = 2 (2 y^{\frac{3}{2}})$

Step 6.2.2.1.2.2

Rewrite the expression.

$\frac{y'}{y^{\frac{1}{2}}} y^{\frac{3}{2}} - \frac{y'}{2 y^{\frac{3}{2}}} (2 y^{\frac{3}{2}}) = 2 (2 y^{\frac{3}{2}})$

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

Factor $y^{\frac{1}{2}}$ out of $y^{\frac{3}{2}}$ .

$\frac{y'}{y^{\frac{1}{2}}} (y^{\frac{1}{2}} y^{\frac{2}{2}}) - \frac{y'}{2 y^{\frac{3}{2}}} (2 y^{\frac{3}{2}}) = 2 (2 y^{\frac{3}{2}})$

Step 6.2.2.1.3.2

Cancel the common factor.

$\frac{y'}{y^{\frac{1}{2}}} (y^{\frac{1}{2}} y^{\frac{2}{2}}) - \frac{y'}{2 y^{\frac{3}{2}}} (2 y^{\frac{3}{2}}) = 2 (2 y^{\frac{3}{2}})$

Step 6.2.2.1.3.3

Rewrite the expression.

$y' y^{\frac{2}{2}} - \frac{y'}{2 y^{\frac{3}{2}}} (2 y^{\frac{3}{2}}) = 2 (2 y^{\frac{3}{2}})$

Step 6.2.2.1.4

Divide $2$ by $2$ .

$y' y^{1} - \frac{y'}{2 y^{\frac{3}{2}}} (2 y^{\frac{3}{2}}) = 2 (2 y^{\frac{3}{2}})$

Step 6.2.2.1.5

Simplify.

$y' y - \frac{y'}{2 y^{\frac{3}{2}}} (2 y^{\frac{3}{2}}) = 2 (2 y^{\frac{3}{2}})$

Step 6.2.2.1.6

Cancel the common factor of $2 y^{\frac{3}{2}}$ .

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Step 6.2.2.1.6.1

Move the leading negative in $- \frac{y'}{2 y^{\frac{3}{2}}}$ into the numerator.

$y' y + \frac{- y'}{2 y^{\frac{3}{2}}} (2 y^{\frac{3}{2}}) = 2 (2 y^{\frac{3}{2}})$

Step 6.2.2.1.6.2

Cancel the common factor.

$y' y + \frac{- y'}{2 y^{\frac{3}{2}}} (2 y^{\frac{3}{2}}) = 2 (2 y^{\frac{3}{2}})$

Step 6.2.2.1.6.3

Rewrite the expression.

$y' y - y' = 2 (2 y^{\frac{3}{2}})$

Step 6.2.3

Simplify the right side.

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Step 6.2.3.1

Multiply $2$ by $2$ .

$y' y - y' = 4 y^{\frac{3}{2}}$

Step 6.3

Solve the equation.

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Step 6.3.1

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

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Step 6.3.1.1

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

$y' (y) - y' = 4 y^{\frac{3}{2}}$

Step 6.3.1.2

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

$y' (y) + y' \cdot - 1 = 4 y^{\frac{3}{2}}$

Step 6.3.1.3

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

$y' (y - 1) = 4 y^{\frac{3}{2}}$

Step 6.3.2

Divide each term in $y' (y - 1) = 4 y^{\frac{3}{2}}$ by $y - 1$ and simplify.

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Step 6.3.2.1

Divide each term in $y' (y - 1) = 4 y^{\frac{3}{2}}$ by $y - 1$ .

$\frac{y' (y - 1)}{y - 1} = \frac{4 y^{\frac{3}{2}}}{y - 1}$

Step 6.3.2.2

Simplify the left side.

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Step 6.3.2.2.1

Cancel the common factor of $y - 1$ .

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Step 6.3.2.2.1.1

Cancel the common factor.

$\frac{y' (y - 1)}{y - 1} = \frac{4 y^{\frac{3}{2}}}{y - 1}$

Step 6.3.2.2.1.2

Divide $y'$ by $1$ .

$y' = \frac{4 y^{\frac{3}{2}}}{y - 1}$

Step 7

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

$\frac{d y}{d x} = \frac{4 y^{\frac{3}{2}}}{y - 1}$