Calculus Examples

$8 \sqrt{y} = x - 3 y$

Step 1

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

$8 y^{\frac{1}{2}} = x - 3 y$

Step 2

Differentiate both sides of the equation.

$\frac{d}{d x} (8 y^{\frac{1}{2}}) = \frac{d}{d x} (x - 3 y)$

Step 3

Differentiate the left side of the equation.

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Step 3.1

Since $8$ is constant with respect to $x$ , the derivative of $8 y^{\frac{1}{2}}$ with respect to $x$ is $8 \frac{d}{d x} [y^{\frac{1}{2}}]$ .

$8 \frac{d}{d x} [y^{\frac{1}{2}}]$

Step 3.2

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

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Step 3.2.1

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

$8 (\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [y])$

Step 3.2.2

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

$8 (\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [y])$

Step 3.2.3

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

$8 (\frac{1}{2} y^{\frac{1}{2} - 1} \frac{d}{d x} [y])$

Step 3.3

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

$8 (\frac{1}{2} y^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [y])$

Step 3.4

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

$8 (\frac{1}{2} y^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [y])$

Step 3.5

Combine the numerators over the common denominator.

$8 (\frac{1}{2} y^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [y])$

Step 3.6

Simplify the numerator.

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Step 3.6.1

Multiply $- 1$ by $2$ .

$8 (\frac{1}{2} y^{\frac{1 - 2}{2}} \frac{d}{d x} [y])$

Step 3.6.2

Subtract $2$ from $1$ .

$8 (\frac{1}{2} y^{\frac{- 1}{2}} \frac{d}{d x} [y])$

Step 3.7

Move the negative in front of the fraction.

$8 (\frac{1}{2} y^{- \frac{1}{2}} \frac{d}{d x} [y])$

Step 3.8

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

$8 (\frac{y^{- \frac{1}{2}}}{2} \frac{d}{d x} [y])$

Step 3.9

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

$8 (\frac{1}{2 y^{\frac{1}{2}}} \frac{d}{d x} [y])$

Step 3.10

Combine $\frac{1}{2 y^{\frac{1}{2}}}$ and $8$ .

$\frac{8}{2 y^{\frac{1}{2}}} \frac{d}{d x} [y]$

Step 3.11

Factor $2$ out of $8$ .

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

Step 3.12

Cancel the common factors.

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Step 3.12.1

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

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

Step 3.12.2

Cancel the common factor.

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

Step 3.12.3

Rewrite the expression.

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

Step 3.13

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

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

Step 3.14

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

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

Step 4

Differentiate the right side of the equation.

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Step 4.1

Differentiate.

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Step 4.1.1

By the Sum Rule, the derivative of $x - 3 y$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 3 y]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [- 3 y]$

Step 4.1.2

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

$1 + \frac{d}{d x} [- 3 y]$

Step 4.2

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

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Step 4.2.1

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

$1 - 3 \frac{d}{d x} [y]$

Step 4.2.2

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

$1 - 3 y'$

Step 5

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

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

Step 6

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

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Step 6.1

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

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

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

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

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Step 6.2.1.1.1

Cancel the common factor.

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

Step 6.2.1.1.2

Rewrite the expression.

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

Step 6.2.2

Simplify the right side.

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Step 6.2.2.1

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

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Step 6.2.2.1.1

Apply the distributive property.

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

Step 6.2.2.1.2

Simplify the expression.

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Step 6.2.2.1.2.1

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

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

Step 6.2.2.1.2.2

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

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

Step 6.3

Solve for $y'$ .

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Step 6.3.1

Add $3 y' y^{\frac{1}{2}}$ to both sides of the equation.

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

Step 6.3.2

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

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

Step 6.3.3

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

$4 {({y'}^{\frac{1}{2}})}^{2} + 3 y' u - (u) = 0$

Step 6.3.4

Solve for $u$ .

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Step 6.3.4.1

Simplify each term.

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Step 6.3.4.1.1

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

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Step 6.3.4.1.1.1

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

$4 {y'}^{\frac{1}{2} \cdot 2} + 3 y' u - u = 0$

Step 6.3.4.1.1.2

Cancel the common factor of $2$ .

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Step 6.3.4.1.1.2.1

Cancel the common factor.

$4 {y'}^{\frac{1}{2} \cdot 2} + 3 y' u - u = 0$

Step 6.3.4.1.1.2.2

Rewrite the expression.

$4 {y'}^{1} + 3 y' u - u = 0$

Step 6.3.4.1.2

Simplify.

$4 y' + 3 y' u - u = 0$

Step 6.3.4.2

Subtract $4 y'$ from both sides of the equation.

$3 y' u - u = - 4 y'$

Step 6.3.4.3

Factor $u$ out of $3 y' u - u$ .

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Step 6.3.4.3.1

Factor $u$ out of $3 y' u$ .

$u (3 y') - u = - 4 y'$

Step 6.3.4.3.2

Factor $u$ out of $- u$ .

$u (3 y') + u \cdot - 1 = - 4 y'$

Step 6.3.4.3.3

Factor $u$ out of $u (3 y') + u \cdot - 1$ .

$u (3 y' - 1) = - 4 y'$

Step 6.3.4.4

Divide each term in $u (3 y' - 1) = - 4 y'$ by $3 y' - 1$ and simplify.

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Step 6.3.4.4.1

Divide each term in $u (3 y' - 1) = - 4 y'$ by $3 y' - 1$ .

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

Step 6.3.4.4.2

Simplify the left side.

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Step 6.3.4.4.2.1

Cancel the common factor of $3 y' - 1$ .

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Step 6.3.4.4.2.1.1

Cancel the common factor.

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

Step 6.3.4.4.2.1.2

Divide $u$ by $1$ .

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

Step 6.3.4.4.3

Simplify the right side.

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Step 6.3.4.4.3.1

Move the negative in front of the fraction.

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

Step 6.3.5

Substitute $y'$ for $u$ .

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

Step 7

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

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