Algebra Examples

$y = x^{0.85}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d y} (y) = \frac{d}{d y} (x^{0.85})$

Step 2

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

$1$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Differentiate using the chain rule, which states that $\frac{d}{d y} [f (g (y))]$ is $f' (g (y)) g' (y)$ where $f (y) = y^{0.85}$ and $g (y) = x$ .

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Step 3.1.1

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

$\frac{d}{d u} [u^{0.85}] \frac{d}{d y} [x]$

Step 3.1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 0.85$ .

$0.85 u^{- 0.15} \frac{d}{d y} [x]$

Step 3.1.3

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

$0.85 x^{- 0.15} \frac{d}{d y} [x]$

Step 3.2

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

$0.85 x^{- 0.15} x'$

Step 3.3

Simplify.

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Step 3.3.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$0.85 \frac{1}{x^{0.15}} x'$

Step 3.3.2

Combine terms.

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Step 3.3.2.1

Combine $0.85$ and $\frac{1}{x^{0.15}}$ .

$\frac{0.85}{x^{0.15}} x'$

Step 3.3.2.2

Combine $\frac{0.85}{x^{0.15}}$ and $x'$ .

$\frac{0.85 x'}{x^{0.15}}$

Step 4

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

$1 = \frac{0.85 x'}{x^{0.15}}$

Step 5

Solve for $x'$ .

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Step 5.1

Rewrite the equation as $\frac{0.85 x'}{x^{0.15}} = 1$ .

$\frac{0.85 x'}{x^{0.15}} = 1$

Step 5.2

Multiply both sides by $x^{0.15}$ .

$\frac{0.85 x'}{x^{0.15}} x^{0.15} = 1 x^{0.15}$

Step 5.3

Simplify.

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Step 5.3.1

Simplify the left side.

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Step 5.3.1.1

Cancel the common factor of $x^{0.15}$ .

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Step 5.3.1.1.1

Cancel the common factor.

$\frac{0.85 x'}{x^{0.15}} x^{0.15} = 1 x^{0.15}$

Step 5.3.1.1.2

Rewrite the expression.

$0.85 x' = 1 x^{0.15}$

Step 5.3.2

Simplify the right side.

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Step 5.3.2.1

Multiply $x^{0.15}$ by $1$ .

$0.85 x' = x^{0.15}$

Step 5.4

Divide each term in $0.85 x' = x^{0.15}$ by $0.85$ and simplify.

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Step 5.4.1

Divide each term in $0.85 x' = x^{0.15}$ by $0.85$ .

$\frac{0.85 x'}{0.85} = \frac{x^{0.15}}{0.85}$

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of $0.85$ .

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Step 5.4.2.1.1

Cancel the common factor.

$\frac{0.85 x'}{0.85} = \frac{x^{0.15}}{0.85}$

Step 5.4.2.1.2

Divide $x'$ by $1$ .

$x' = \frac{x^{0.15}}{0.85}$

Step 5.4.3

Simplify the right side.

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Step 5.4.3.1

Multiply by $1$ .

$x' = \frac{1 x^{0.15}}{0.85}$

Step 5.4.3.2

Factor $0.85$ out of $0.85$ .

$x' = \frac{1 x^{0.15}}{0.85 (1)}$

Step 5.4.3.3

Separate fractions.

$x' = \frac{1}{0.85} \cdot \frac{x^{0.15}}{1}$

Step 5.4.3.4

Divide $1$ by $0.85$ .

$x' = 1.17647058 \frac{x^{0.15}}{1}$

Step 5.4.3.5

Divide $x^{0.15}$ by $1$ .

$x' = 1.17647058 x^{0.15}$

Step 6

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

$\frac{d x}{d y} = 1.17647058 x^{0.15}$