Calculus Examples

$w = \frac{2}{3} - \frac{y^{2}}{5} + \frac{4 y^{2}}{3}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d w} (w) = \frac{d}{d w} (\frac{2}{3} - \frac{y^{2}}{5} + \frac{4 y^{2}}{3})$

Step 2

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

$1$

Step 3

Differentiate the right 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 $\frac{2}{3} - \frac{y^{2}}{5} + \frac{4 y^{2}}{3}$ with respect to $w$ is $\frac{d}{d w} [\frac{2}{3}] + \frac{d}{d w} [- \frac{y^{2}}{5}] + \frac{d}{d w} [\frac{4 y^{2}}{3}]$ .

$\frac{d}{d w} [\frac{2}{3}] + \frac{d}{d w} [- \frac{y^{2}}{5}] + \frac{d}{d w} [\frac{4 y^{2}}{3}]$

Step 3.1.2

Since $\frac{2}{3}$ is constant with respect to $w$ , the derivative of $\frac{2}{3}$ with respect to $w$ is $0$ .

$0 + \frac{d}{d w} [- \frac{y^{2}}{5}] + \frac{d}{d w} [\frac{4 y^{2}}{3}]$

Step 3.2

Evaluate $\frac{d}{d w} [- \frac{y^{2}}{5}]$ .

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Step 3.2.1

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

$0 - \frac{1}{5} \frac{d}{d w} [y^{2}] + \frac{d}{d w} [\frac{4 y^{2}}{3}]$

Step 3.2.2

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

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Step 3.2.2.1

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

$0 - \frac{1}{5} (\frac{d}{d u_{1}} [{u_{1}}^{2}] \frac{d}{d w} [y]) + \frac{d}{d w} [\frac{4 y^{2}}{3}]$

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 $n = 2$ .

$0 - \frac{1}{5} (2 u_{1} \frac{d}{d w} [y]) + \frac{d}{d w} [\frac{4 y^{2}}{3}]$

Step 3.2.2.3

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

$0 - \frac{1}{5} (2 y \frac{d}{d w} [y]) + \frac{d}{d w} [\frac{4 y^{2}}{3}]$

Step 3.2.3

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

$0 - \frac{1}{5} (2 y y') + \frac{d}{d w} [\frac{4 y^{2}}{3}]$

Step 3.2.4

Multiply $2$ by $- 1$ .

$0 - 2 (\frac{1}{5}) (y y') + \frac{d}{d w} [\frac{4 y^{2}}{3}]$

Step 3.2.5

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

$0 + \frac{- 2}{5} (y y') + \frac{d}{d w} [\frac{4 y^{2}}{3}]$

Step 3.2.6

Combine $y$ and $\frac{- 2}{5}$ .

$0 + \frac{y \cdot - 2}{5} y' + \frac{d}{d w} [\frac{4 y^{2}}{3}]$

Step 3.2.7

Combine $\frac{y \cdot - 2}{5}$ and $y'$ .

$0 + \frac{y \cdot - 2 y'}{5} + \frac{d}{d w} [\frac{4 y^{2}}{3}]$

Step 3.2.8

Move $- 2$ to the left of $y$ .

$0 + \frac{- 2 \cdot y y'}{5} + \frac{d}{d w} [\frac{4 y^{2}}{3}]$

Step 3.2.9

Move the negative in front of the fraction.

$0 - \frac{2 y y'}{5} + \frac{d}{d w} [\frac{4 y^{2}}{3}]$

Step 3.3

Evaluate $\frac{d}{d w} [\frac{4 y^{2}}{3}]$ .

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Step 3.3.1

Since $\frac{4}{3}$ is constant with respect to $w$ , the derivative of $\frac{4 y^{2}}{3}$ with respect to $w$ is $\frac{4}{3} \frac{d}{d w} [y^{2}]$ .

$0 - \frac{2 y y'}{5} + \frac{4}{3} \frac{d}{d w} [y^{2}]$

Step 3.3.2

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

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Step 3.3.2.1

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

$0 - \frac{2 y y'}{5} + \frac{4}{3} (\frac{d}{d u_{2}} [{u_{2}}^{2}] \frac{d}{d w} [y])$

Step 3.3.2.2

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

$0 - \frac{2 y y'}{5} + \frac{4}{3} (2 u_{2} \frac{d}{d w} [y])$

Step 3.3.2.3

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

$0 - \frac{2 y y'}{5} + \frac{4}{3} (2 y \frac{d}{d w} [y])$

Step 3.3.3

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

$0 - \frac{2 y y'}{5} + \frac{4}{3} (2 y y')$

Step 3.3.4

Combine $2$ and $\frac{4}{3}$ .

$0 - \frac{2 y y'}{5} + \frac{2 \cdot 4}{3} (y y')$

Step 3.3.5

Multiply $2$ by $4$ .

$0 - \frac{2 y y'}{5} + \frac{8}{3} (y y')$

Step 3.3.6

Combine $y$ and $\frac{8}{3}$ .

$0 - \frac{2 y y'}{5} + \frac{y \cdot 8}{3} y'$

Step 3.3.7

Combine $\frac{y \cdot 8}{3}$ and $y'$ .

$0 - \frac{2 y y'}{5} + \frac{y \cdot 8 y'}{3}$

Step 3.3.8

Move $8$ to the left of $y$ .

$0 - \frac{2 y y'}{5} + \frac{8 y y'}{3}$

Step 3.4

Combine terms.

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Step 3.4.1

Subtract $\frac{2 y y'}{5}$ from $0$ .

$- \frac{2 y y'}{5} + \frac{8 y y'}{3}$

Step 3.4.2

To write $- \frac{2 y y'}{5}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{2 y y'}{5} \cdot \frac{3}{3} + \frac{8 y y'}{3}$

Step 3.4.3

To write $\frac{8 y y'}{3}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$- \frac{2 y y'}{5} \cdot \frac{3}{3} + \frac{8 y y'}{3} \cdot \frac{5}{5}$

Step 3.4.4

Write each expression with a common denominator of $15$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.4.4.1

Multiply $\frac{2 y y'}{5}$ by $\frac{3}{3}$ .

$- \frac{2 y y' \cdot 3}{5 \cdot 3} + \frac{8 y y'}{3} \cdot \frac{5}{5}$

Step 3.4.4.2

Multiply $5$ by $3$ .

$- \frac{2 y y' \cdot 3}{15} + \frac{8 y y'}{3} \cdot \frac{5}{5}$

Step 3.4.4.3

Multiply $\frac{8 y y'}{3}$ by $\frac{5}{5}$ .

$- \frac{2 y y' \cdot 3}{15} + \frac{8 y y' \cdot 5}{3 \cdot 5}$

Step 3.4.4.4

Multiply $3$ by $5$ .

$- \frac{2 y y' \cdot 3}{15} + \frac{8 y y' \cdot 5}{15}$

Step 3.4.5

Combine the numerators over the common denominator.

$\frac{- 2 y y' \cdot 3 + 8 y y' \cdot 5}{15}$

Step 3.4.6

Multiply $3$ by $- 2$ .

$\frac{- 6 y y' + 8 y y' \cdot 5}{15}$

Step 3.4.7

Multiply $5$ by $8$ .

$\frac{- 6 y y' + 40 y y'}{15}$

Step 3.4.8

Add $- 6 y y'$ and $40 y y'$ .

$\frac{34 y y'}{15}$

Step 4

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

$1 = \frac{34 y y'}{15}$

Step 5

Solve for $y'$ .

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Step 5.1

Rewrite the equation as $\frac{34 y y'}{15} = 1$ .

$\frac{34 y y'}{15} = 1$

Step 5.2

Multiply both sides of the equation by $\frac{15}{34}$ .

$\frac{15}{34} \cdot \frac{34 y y'}{15} = \frac{15}{34} \cdot 1$

Step 5.3

Simplify both sides of the equation.

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Step 5.3.1

Simplify the left side.

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Step 5.3.1.1

Simplify $\frac{15}{34} \cdot \frac{34 y y'}{15}$ .

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Step 5.3.1.1.1

Cancel the common factor of $15$ .

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Step 5.3.1.1.1.1

Cancel the common factor.

$\frac{15}{34} \cdot \frac{34 y y'}{15} = \frac{15}{34} \cdot 1$

Step 5.3.1.1.1.2

Rewrite the expression.

$\frac{1}{34} (34 y y') = \frac{15}{34} \cdot 1$

Step 5.3.1.1.2

Cancel the common factor of $34$ .

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Step 5.3.1.1.2.1

Factor $34$ out of $34 y y'$ .

$\frac{1}{34} (34 (y y')) = \frac{15}{34} \cdot 1$

Step 5.3.1.1.2.2

Cancel the common factor.

$\frac{1}{34} (34 (y y')) = \frac{15}{34} \cdot 1$

Step 5.3.1.1.2.3

Rewrite the expression.

$y y' = \frac{15}{34} \cdot 1$

Step 5.3.2

Simplify the right side.

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Step 5.3.2.1

Multiply $\frac{15}{34}$ by $1$ .

$y y' = \frac{15}{34}$

Step 5.4

Divide each term in $y y' = \frac{15}{34}$ by $y$ and simplify.

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Step 5.4.1

Divide each term in $y y' = \frac{15}{34}$ by $y$ .

$\frac{y y'}{y} = \frac{\frac{15}{34}}{y}$

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of $y$ .

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Step 5.4.2.1.1

Cancel the common factor.

$\frac{y y'}{y} = \frac{\frac{15}{34}}{y}$

Step 5.4.2.1.2

Divide $y'$ by $1$ .

$y' = \frac{\frac{15}{34}}{y}$

Step 5.4.3

Simplify the right side.

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Step 5.4.3.1

Multiply the numerator by the reciprocal of the denominator.

$y' = \frac{15}{34} \cdot \frac{1}{y}$

Step 5.4.3.2

Multiply $\frac{15}{34}$ by $\frac{1}{y}$ .

$y' = \frac{15}{34 y}$

Step 6

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

$\frac{d y}{d w} = \frac{15}{34 y}$