Calculus Examples

$v (y) = x - \frac{3}{95} x^{3}$

Step 1

Multiply $y'$ by $y$ .

$y' y = x - \frac{3}{95} x^{3}$

Step 2

Differentiate both sides of the equation.

$\frac{d}{d x} (y' y) = \frac{d}{d x} (x - \frac{3}{95} x^{3})$

Step 3

Differentiate the left side of the equation.

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Step 3.1

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

$y' \frac{d}{d x} [y]$

Step 3.2

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

$y' y'$

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 $x - \frac{3}{95} x^{3}$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- \frac{3}{95} x^{3}]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [- \frac{3}{95} x^{3}]$

Step 4.1.2

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

$1 + \frac{d}{d x} [- \frac{3}{95} x^{3}]$

Step 4.2

Evaluate $\frac{d}{d x} [- \frac{3}{95} x^{3}]$ .

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Step 4.2.1

Since $- \frac{3}{95}$ is constant with respect to $x$ , the derivative of $- \frac{3}{95} x^{3}$ with respect to $x$ is $- \frac{3}{95} \frac{d}{d x} [x^{3}]$ .

$1 - \frac{3}{95} \frac{d}{d x} [x^{3}]$

Step 4.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$1 - \frac{3}{95} (3 x^{2})$

Step 4.2.3

Multiply $3$ by $- 1$ .

$1 - 3 (\frac{3}{95}) x^{2}$

Step 4.2.4

Combine $- 3$ and $\frac{3}{95}$ .

$1 + \frac{- 3 \cdot 3}{95} x^{2}$

Step 4.2.5

Multiply $- 3$ by $3$ .

$1 + \frac{- 9}{95} x^{2}$

Step 4.2.6

Combine $\frac{- 9}{95}$ and $x^{2}$ .

$1 + \frac{- 9 x^{2}}{95}$

Step 4.2.7

Move the negative in front of the fraction.

$1 - \frac{9 x^{2}}{95}$

Step 4.3

Reorder terms.

$- \frac{9 x^{2}}{95} + 1$

Step 5

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

$y' y' = - \frac{9 x^{2}}{95} + 1$

Step 6

Solve for $y'$ .

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Step 6.1

Multiply $y'$ by $y'$ .

${y'}^{2} = - \frac{9 x^{2}}{95} + 1$

Step 6.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$y' = \pm \sqrt{- \frac{9 x^{2}}{95} + 1}$

Step 6.3

Simplify $\pm \sqrt{- \frac{9 x^{2}}{95} + 1}$ .

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Step 6.3.1

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

$y' = \pm \sqrt{- \frac{9 x^{2}}{95} + \frac{95}{95}}$

Step 6.3.2

Combine the numerators over the common denominator.

$y' = \pm \sqrt{\frac{- 9 x^{2} + 95}{95}}$

Step 6.3.3

Rewrite $\sqrt{\frac{- 9 x^{2} + 95}{95}}$ as $\frac{\sqrt{- 9 x^{2} + 95}}{\sqrt{95}}$ .

$y' = \pm \frac{\sqrt{- 9 x^{2} + 95}}{\sqrt{95}}$

Step 6.3.4

Multiply $\frac{\sqrt{- 9 x^{2} + 95}}{\sqrt{95}}$ by $\frac{\sqrt{95}}{\sqrt{95}}$ .

$y' = \pm \frac{\sqrt{- 9 x^{2} + 95}}{\sqrt{95}} \cdot \frac{\sqrt{95}}{\sqrt{95}}$

Step 6.3.5

Combine and simplify the denominator.

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Step 6.3.5.1

Multiply $\frac{\sqrt{- 9 x^{2} + 95}}{\sqrt{95}}$ by $\frac{\sqrt{95}}{\sqrt{95}}$ .

$y' = \pm \frac{\sqrt{- 9 x^{2} + 95} \sqrt{95}}{\sqrt{95} \sqrt{95}}$

Step 6.3.5.2

Raise $\sqrt{95}$ to the power of $1$ .

$y' = \pm \frac{\sqrt{- 9 x^{2} + 95} \sqrt{95}}{{\sqrt{95}}^{1} \sqrt{95}}$

Step 6.3.5.3

Raise $\sqrt{95}$ to the power of $1$ .

$y' = \pm \frac{\sqrt{- 9 x^{2} + 95} \sqrt{95}}{{\sqrt{95}}^{1} {\sqrt{95}}^{1}}$

Step 6.3.5.4

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

$y' = \pm \frac{\sqrt{- 9 x^{2} + 95} \sqrt{95}}{{\sqrt{95}}^{1 + 1}}$

Step 6.3.5.5

Add $1$ and $1$ .

$y' = \pm \frac{\sqrt{- 9 x^{2} + 95} \sqrt{95}}{{\sqrt{95}}^{2}}$

Step 6.3.5.6

Rewrite ${\sqrt{95}}^{2}$ as $95$ .

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Step 6.3.5.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{95}$ as $95^{\frac{1}{2}}$ .

$y' = \pm \frac{\sqrt{- 9 x^{2} + 95} \sqrt{95}}{{(95^{\frac{1}{2}})}^{2}}$

Step 6.3.5.6.2

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

$y' = \pm \frac{\sqrt{- 9 x^{2} + 95} \sqrt{95}}{95^{\frac{1}{2} \cdot 2}}$

Step 6.3.5.6.3

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

$y' = \pm \frac{\sqrt{- 9 x^{2} + 95} \sqrt{95}}{95^{\frac{2}{2}}}$

Step 6.3.5.6.4

Cancel the common factor of $2$ .

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Step 6.3.5.6.4.1

Cancel the common factor.

$y' = \pm \frac{\sqrt{- 9 x^{2} + 95} \sqrt{95}}{95^{\frac{2}{2}}}$

Step 6.3.5.6.4.2

Rewrite the expression.

$y' = \pm \frac{\sqrt{- 9 x^{2} + 95} \sqrt{95}}{95^{1}}$

Step 6.3.5.6.5

Evaluate the exponent.

$y' = \pm \frac{\sqrt{- 9 x^{2} + 95} \sqrt{95}}{95}$

Step 6.3.6

Combine using the product rule for radicals.

$y' = \pm \frac{\sqrt{(- 9 x^{2} + 95) \cdot 95}}{95}$

Step 6.3.7

Reorder factors in $\pm \frac{\sqrt{(- 9 x^{2} + 95) \cdot 95}}{95}$ .

$y' = \pm \frac{\sqrt{95 (- 9 x^{2} + 95)}}{95}$

Step 6.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 6.4.1

First, use the positive value of the $\pm$ to find the first solution.

$y' = \frac{\sqrt{95 (- 9 x^{2} + 95)}}{95}$

Step 6.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$y' = - \frac{\sqrt{95 (- 9 x^{2} + 95)}}{95}$

Step 6.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$y' = \frac{\sqrt{95 (- 9 x^{2} + 95)}}{95}, - \frac{\sqrt{95 (- 9 x^{2} + 95)}}{95}$

Step 7

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

$\frac{d y}{d x} = \frac{\sqrt{95 (- 9 x^{2} + 95)}}{95}, - \frac{\sqrt{95 (- 9 x^{2} + 95)}}{95}$