Calculus Examples

$4 \sqrt{y} - y = 2 x$

Step 1

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

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

Step 2

Differentiate both sides of the equation.

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

Step 3

Differentiate the left side of the equation.

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Step 3.1

By the Sum Rule, the derivative of $4 y^{\frac{1}{2}} - y$ with respect to $x$ is $\frac{d}{d x} [4 y^{\frac{1}{2}}] + \frac{d}{d x} [- y]$ .

$\frac{d}{d x} [4 y^{\frac{1}{2}}] + \frac{d}{d x} [- y]$

Step 3.2

Evaluate $\frac{d}{d x} [4 y^{\frac{1}{2}}]$ .

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Step 3.2.1

Since $4$ is constant with respect to $x$ , the derivative of $4 y^{\frac{1}{2}}$ with respect to $x$ is $4 \frac{d}{d x} [y^{\frac{1}{2}}]$ .

$4 \frac{d}{d x} [y^{\frac{1}{2}}] + \frac{d}{d x} [- y]$

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

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Step 3.2.2.1

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

$4 (\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [y]) + \frac{d}{d x} [- y]$

Step 3.2.2.2

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

$4 (\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [y]) + \frac{d}{d x} [- y]$

Step 3.2.2.3

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

$4 (\frac{1}{2} y^{\frac{1}{2} - 1} \frac{d}{d x} [y]) + \frac{d}{d x} [- y]$

Step 3.2.3

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

$4 (\frac{1}{2} y^{\frac{1}{2} - 1} y') + \frac{d}{d x} [- y]$

Step 3.2.4

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

$4 (\frac{1}{2} y^{\frac{1}{2} - 1 \cdot \frac{2}{2}} y') + \frac{d}{d x} [- y]$

Step 3.2.5

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

$4 (\frac{1}{2} y^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} y') + \frac{d}{d x} [- y]$

Step 3.2.6

Combine the numerators over the common denominator.

$4 (\frac{1}{2} y^{\frac{1 - 1 \cdot 2}{2}} y') + \frac{d}{d x} [- y]$

Step 3.2.7

Simplify the numerator.

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Step 3.2.7.1

Multiply $- 1$ by $2$ .

$4 (\frac{1}{2} y^{\frac{1 - 2}{2}} y') + \frac{d}{d x} [- y]$

Step 3.2.7.2

Subtract $2$ from $1$ .

$4 (\frac{1}{2} y^{\frac{- 1}{2}} y') + \frac{d}{d x} [- y]$

Step 3.2.8

Move the negative in front of the fraction.

$4 (\frac{1}{2} y^{- \frac{1}{2}} y') + \frac{d}{d x} [- y]$

Step 3.2.9

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

$4 (\frac{y^{- \frac{1}{2}}}{2} y') + \frac{d}{d x} [- y]$

Step 3.2.10

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

$4 \frac{y^{- \frac{1}{2}} y'}{2} + \frac{d}{d x} [- y]$

Step 3.2.11

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

$4 \frac{y'}{2 y^{\frac{1}{2}}} + \frac{d}{d x} [- y]$

Step 3.2.12

Combine $4$ and $\frac{y'}{2 y^{\frac{1}{2}}}$ .

$\frac{4 y'}{2 y^{\frac{1}{2}}} + \frac{d}{d x} [- y]$

Step 3.2.13

Factor $2$ out of $4 y'$ .

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

Step 3.2.14

Cancel the common factors.

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Step 3.2.14.1

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

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

Step 3.2.14.2

Cancel the common factor.

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

Step 3.2.14.3

Rewrite the expression.

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

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

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

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

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.

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

Step 6

Solve for $y^{\frac{1}{2}}$ .

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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.

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

Step 6.1.2

The LCM of one and any expression is the expression.

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

Step 6.2

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

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Step 6.2.1

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

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

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

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

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Step 6.2.2.1.1

Cancel the common factor.

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

Step 6.2.2.1.2

Rewrite the expression.

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

Step 6.3

Solve the equation.

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Step 6.3.1

Find a common factor $y^{\frac{1}{2}}$ that is present in each term.

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

Step 6.3.2

Substitute $u$ for $y^{\frac{1}{2}}$ .

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

Step 6.3.3

Solve for $u$ .

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Step 6.3.3.1

Simplify each term.

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Step 6.3.3.1.1

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

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Step 6.3.3.1.1.1

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

$2 {y'}^{\frac{1}{2} \cdot 2} - y' u - 2 u = 0$

Step 6.3.3.1.1.2

Cancel the common factor of $2$ .

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Step 6.3.3.1.1.2.1

Cancel the common factor.

$2 {y'}^{\frac{1}{2} \cdot 2} - y' u - 2 u = 0$

Step 6.3.3.1.1.2.2

Rewrite the expression.

$2 {y'}^{1} - y' u - 2 u = 0$

Step 6.3.3.1.2

Simplify.

$2 y' - y' u - 2 u = 0$

Step 6.3.3.2

Subtract $2 y'$ from both sides of the equation.

$- y' u - 2 u = - 2 y'$

Step 6.3.3.3

Factor $u$ out of $- y' u - 2 u$ .

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Step 6.3.3.3.1

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

$u (- y') - 2 u = - 2 y'$

Step 6.3.3.3.2

Factor $u$ out of $- 2 u$ .

$u (- y') + u \cdot - 2 = - 2 y'$

Step 6.3.3.3.3

Factor $u$ out of $u (- y') + u \cdot - 2$ .

$u (- y' - 2) = - 2 y'$

Step 6.3.3.4

Divide each term in $u (- y' - 2) = - 2 y'$ by $- y' - 2$ and simplify.

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Step 6.3.3.4.1

Divide each term in $u (- y' - 2) = - 2 y'$ by $- y' - 2$ .

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

Step 6.3.3.4.2

Simplify the left side.

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Step 6.3.3.4.2.1

Cancel the common factor of $- y' - 2$ .

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Step 6.3.3.4.2.1.1

Cancel the common factor.

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

Step 6.3.3.4.2.1.2

Divide $u$ by $1$ .

$u = \frac{- 2 y'}{- y' - 2}$

Step 6.3.3.4.3

Simplify the right side.

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Step 6.3.3.4.3.1

Move the negative in front of the fraction.

$u = - \frac{2 y'}{- y' - 2}$

Step 6.3.3.4.3.2

Factor $- 1$ out of $- y'$ .

$u = - \frac{2 y'}{- y' - 2}$

Step 6.3.3.4.3.3

Rewrite $- 2$ as $- 1 (2)$ .

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

Step 6.3.3.4.3.4

Factor $- 1$ out of $- y' - 1 (2)$ .

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

Step 6.3.3.4.3.5

Simplify the expression.

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Step 6.3.3.4.3.5.1

Rewrite $- (y' + 2)$ as $- 1 (y' + 2)$ .

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

Step 6.3.3.4.3.5.2

Move the negative in front of the fraction.

$u = - - \frac{2 y'}{y' + 2}$

Step 6.3.3.4.3.5.3

Multiply $- 1$ by $- 1$ .

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

Step 6.3.3.4.3.5.4

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

$u = \frac{2 y'}{y' + 2}$

Step 6.3.4

Substitute $y'$ for $u$ .

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

Step 7

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

$\frac{d y}{d x} = \frac{2 y'}{y' + 2}$