Calculus Examples

$a^{3} y = x^{2} (2 a^{2} - x^{2})$

Step 1

Differentiate both sides of the equation.

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

Step 2

Differentiate the left side of the equation.

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Step 2.1

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

$a^{3} \frac{d}{d x} [y]$

Step 2.2

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

$a^{3} y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x^{2}$ and $g (x) = 2 a^{2} - x^{2}$ .

$x^{2} \frac{d}{d x} [2 a^{2} - x^{2}] + (2 a^{2} - x^{2}) \frac{d}{d x} [x^{2}]$

Step 3.2

Differentiate.

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Step 3.2.1

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

$x^{2} (\frac{d}{d x} [2 a^{2}] + \frac{d}{d x} [- x^{2}]) + (2 a^{2} - x^{2}) \frac{d}{d x} [x^{2}]$

Step 3.2.2

Since $2 a^{2}$ is constant with respect to $x$ , the derivative of $2 a^{2}$ with respect to $x$ is $0$ .

$x^{2} (0 + \frac{d}{d x} [- x^{2}]) + (2 a^{2} - x^{2}) \frac{d}{d x} [x^{2}]$

Step 3.2.3

Add $0$ and $\frac{d}{d x} [- x^{2}]$ .

$x^{2} \frac{d}{d x} [- x^{2}] + (2 a^{2} - x^{2}) \frac{d}{d x} [x^{2}]$

Step 3.2.4

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

$x^{2} (- \frac{d}{d x} [x^{2}]) + (2 a^{2} - x^{2}) \frac{d}{d x} [x^{2}]$

Step 3.2.5

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

$x^{2} (- (2 x)) + (2 a^{2} - x^{2}) \frac{d}{d x} [x^{2}]$

Step 3.2.6

Multiply $2$ by $- 1$ .

$x^{2} (- 2 x) + (2 a^{2} - x^{2}) \frac{d}{d x} [x^{2}]$

Step 3.3

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 3.3.1

Move $x$ .

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

Step 3.3.2

Multiply $x$ by $x^{2}$ .

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Step 3.3.2.1

Raise $x$ to the power of $1$ .

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

Step 3.3.2.2

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

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

Step 3.3.3

Add $1$ and $2$ .

$x^{3} \cdot - 2 + (2 a^{2} - x^{2}) \frac{d}{d x} [x^{2}]$

Step 3.4

Move $- 2$ to the left of $x^{3}$ .

$- 2 \cdot x^{3} + (2 a^{2} - x^{2}) \frac{d}{d x} [x^{2}]$

Step 3.5

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

$- 2 x^{3} + (2 a^{2} - x^{2}) (2 x)$

Step 3.6

Move $2$ to the left of $2 a^{2} - x^{2}$ .

$- 2 x^{3} + 2 \cdot (2 a^{2} - x^{2}) x$

Step 3.7

Simplify.

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Step 3.7.1

Apply the distributive property.

$- 2 x^{3} + (2 (2 a^{2}) + 2 (- x^{2})) x$

Step 3.7.2

Apply the distributive property.

$- 2 x^{3} + 2 (2 a^{2}) x + 2 (- x^{2}) x$

Step 3.7.3

Combine terms.

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Step 3.7.3.1

Multiply $2$ by $2$ .

$- 2 x^{3} + 4 a^{2} x + 2 (- x^{2}) x$

Step 3.7.3.2

Multiply $- 1$ by $2$ .

$- 2 x^{3} + 4 a^{2} x - 2 x^{2} x$

Step 3.7.3.3

Raise $x$ to the power of $1$ .

$- 2 x^{3} + 4 a^{2} x - 2 (x^{1} x^{2})$

Step 3.7.3.4

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

$- 2 x^{3} + 4 a^{2} x - 2 x^{1 + 2}$

Step 3.7.3.5

Add $1$ and $2$ .

$- 2 x^{3} + 4 a^{2} x - 2 x^{3}$

Step 3.7.3.6

Subtract $2 x^{3}$ from $- 2 x^{3}$ .

$- 4 x^{3} + 4 a^{2} x$

Step 4

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

$a^{3} y' = - 4 x^{3} + 4 a^{2} x$

Step 5

Divide each term in $a^{3} y' = - 4 x^{3} + 4 a^{2} x$ by $a^{3}$ and simplify.

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Step 5.1

Divide each term in $a^{3} y' = - 4 x^{3} + 4 a^{2} x$ by $a^{3}$ .

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

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $a^{3}$ .

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Step 5.2.1.1

Cancel the common factor.

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

Step 5.2.1.2

Divide $y'$ by $1$ .

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

Step 5.3

Simplify the right side.

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Step 5.3.1

Simplify each term.

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Step 5.3.1.1

Move the negative in front of the fraction.

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

Step 5.3.1.2

Cancel the common factor of $a^{2}$ and $a^{3}$ .

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Step 5.3.1.2.1

Factor $a^{2}$ out of $4 a^{2} x$ .

$y' = - \frac{4 x^{3}}{a^{3}} + \frac{a^{2} (4 x)}{a^{3}}$

Step 5.3.1.2.2

Cancel the common factors.

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Step 5.3.1.2.2.1

Factor $a^{2}$ out of $a^{3}$ .

$y' = - \frac{4 x^{3}}{a^{3}} + \frac{a^{2} (4 x)}{a^{2} a}$

Step 5.3.1.2.2.2

Cancel the common factor.

$y' = - \frac{4 x^{3}}{a^{3}} + \frac{a^{2} (4 x)}{a^{2} a}$

Step 5.3.1.2.2.3

Rewrite the expression.

$y' = - \frac{4 x^{3}}{a^{3}} + \frac{4 x}{a}$

Step 6

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

$\frac{d y}{d x} = - \frac{4 x^{3}}{a^{3}} + \frac{4 x}{a}$