Calculus Examples

$y^{2} = \frac{x^{2}}{x y - 2500}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y^{2}) = \frac{d}{d x} (\frac{x^{2}}{x y - 2500})$

Step 2

Differentiate the left side of the equation.

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Step 2.1

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

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Step 2.1.1

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

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

Step 2.1.2

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

$2 u \frac{d}{d x} [y]$

Step 2.1.3

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

$2 y \frac{d}{d x} [y]$

Step 2.2

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

$2 y y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = x^{2}$ and $g (x) = x y - 2500$ .

$\frac{(x y - 2500) \frac{d}{d x} [x^{2}] - x^{2} \frac{d}{d x} [x y - 2500]}{{(x y - 2500)}^{2}}$

Step 3.2

Differentiate.

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Step 3.2.1

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

$\frac{(x y - 2500) (2 x) - x^{2} \frac{d}{d x} [x y - 2500]}{{(x y - 2500)}^{2}}$

Step 3.2.2

Move $2$ to the left of $x y - 2500$ .

$\frac{2 \cdot (x y - 2500) x - x^{2} \frac{d}{d x} [x y - 2500]}{{(x y - 2500)}^{2}}$

Step 3.2.3

By the Sum Rule, the derivative of $x y - 2500$ with respect to $x$ is $\frac{d}{d x} [x y] + \frac{d}{d x} [- 2500]$ .

$\frac{2 (x y - 2500) x - x^{2} (\frac{d}{d x} [x y] + \frac{d}{d x} [- 2500])}{{(x y - 2500)}^{2}}$

Step 3.3

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 $f (x) = x$ and $g (x) = y$ .

$\frac{2 (x y - 2500) x - x^{2} (x \frac{d}{d x} [y] + y \frac{d}{d x} [x] + \frac{d}{d x} [- 2500])}{{(x y - 2500)}^{2}}$

Step 3.4

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

$\frac{2 (x y - 2500) x - x^{2} (x y' + y \frac{d}{d x} [x] + \frac{d}{d x} [- 2500])}{{(x y - 2500)}^{2}}$

Step 3.5

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

$\frac{2 (x y - 2500) x - x^{2} (x y' + y \cdot 1 + \frac{d}{d x} [- 2500])}{{(x y - 2500)}^{2}}$

Step 3.6

Multiply $y$ by $1$ .

$\frac{2 (x y - 2500) x - x^{2} (x y' + y + \frac{d}{d x} [- 2500])}{{(x y - 2500)}^{2}}$

Step 3.7

Since $- 2500$ is constant with respect to $x$ , the derivative of $- 2500$ with respect to $x$ is $0$ .

$\frac{2 (x y - 2500) x - x^{2} (x y' + y + 0)}{{(x y - 2500)}^{2}}$

Step 3.8

Add $x y' + y$ and $0$ .

$\frac{2 (x y - 2500) x - x^{2} (x y' + y)}{{(x y - 2500)}^{2}}$

Step 3.9

Simplify.

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Step 3.9.1

Apply the distributive property.

$\frac{(2 (x y) + 2 \cdot - 2500) x - x^{2} (x y' + y)}{{(x y - 2500)}^{2}}$

Step 3.9.2

Apply the distributive property.

$\frac{2 (x y) x + 2 \cdot - 2500 x - x^{2} (x y' + y)}{{(x y - 2500)}^{2}}$

Step 3.9.3

Apply the distributive property.

$\frac{2 (x y) x + 2 \cdot - 2500 x - x^{2} (x y') - x^{2} y}{{(x y - 2500)}^{2}}$

Step 3.9.4

Simplify the numerator.

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Step 3.9.4.1

Simplify each term.

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Step 3.9.4.1.1

Multiply $x$ by $x$ by adding the exponents.

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Step 3.9.4.1.1.1

Move $x$ .

$\frac{2 (x \cdot x y) + 2 \cdot - 2500 x - x^{2} (x y') - x^{2} y}{{(x y - 2500)}^{2}}$

Step 3.9.4.1.1.2

Multiply $x$ by $x$ .

$\frac{2 (x^{2} y) + 2 \cdot - 2500 x - x^{2} (x y') - x^{2} y}{{(x y - 2500)}^{2}}$

Step 3.9.4.1.2

Multiply $2$ by $- 2500$ .

$\frac{2 x^{2} y - 5000 x - x^{2} (x y') - x^{2} y}{{(x y - 2500)}^{2}}$

Step 3.9.4.1.3

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

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Step 3.9.4.1.3.1

Move $x$ .

$\frac{2 x^{2} y - 5000 x - (x \cdot x^{2}) y' - x^{2} y}{{(x y - 2500)}^{2}}$

Step 3.9.4.1.3.2

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

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Step 3.9.4.1.3.2.1

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

$\frac{2 x^{2} y - 5000 x - (x^{1} x^{2}) y' - x^{2} y}{{(x y - 2500)}^{2}}$

Step 3.9.4.1.3.2.2

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

$\frac{2 x^{2} y - 5000 x - x^{1 + 2} y' - x^{2} y}{{(x y - 2500)}^{2}}$

Step 3.9.4.1.3.3

Add $1$ and $2$ .

$\frac{2 x^{2} y - 5000 x - x^{3} y' - x^{2} y}{{(x y - 2500)}^{2}}$

Step 3.9.4.2

Subtract $x^{2} y$ from $2 x^{2} y$ .

$\frac{1 x^{2} y - 5000 x - x^{3} y'}{{(x y - 2500)}^{2}}$

Step 3.9.4.3

Multiply $x^{2}$ by $1$ .

$\frac{x^{2} y - 5000 x - x^{3} y'}{{(x y - 2500)}^{2}}$

Step 3.9.5

Reorder terms.

$\frac{x^{2} y - x^{3} y' - 5000 x}{{(x y - 2500)}^{2}}$

Step 3.9.6

Factor $x$ out of $x^{2} y - x^{3} y' - 5000 x$ .

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Step 3.9.6.1

Factor $x$ out of $x^{2} y$ .

$\frac{x (x y) - x^{3} y' - 5000 x}{{(x y - 2500)}^{2}}$

Step 3.9.6.2

Factor $x$ out of $- x^{3} y'$ .

$\frac{x (x y) + x (- x^{2} y') - 5000 x}{{(x y - 2500)}^{2}}$

Step 3.9.6.3

Factor $x$ out of $- 5000 x$ .

$\frac{x (x y) + x (- x^{2} y') + x \cdot - 5000}{{(x y - 2500)}^{2}}$

Step 3.9.6.4

Factor $x$ out of $x (x y) + x (- x^{2} y')$ .

$\frac{x (x y - x^{2} y') + x \cdot - 5000}{{(x y - 2500)}^{2}}$

Step 3.9.6.5

Factor $x$ out of $x (x y - x^{2} y') + x \cdot - 5000$ .

$\frac{x (x y - x^{2} y' - 5000)}{{(x y - 2500)}^{2}}$

Step 4

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

$2 y y' = \frac{x (x y - x^{2} y' - 5000)}{{(x y - 2500)}^{2}}$

Step 5

Solve for $y'$ .

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Step 5.1

Multiply both sides by ${(x y - 2500)}^{2}$ .

$2 y y' {(x y - 2500)}^{2} = \frac{x (x y - x^{2} y' - 5000)}{{(x y - 2500)}^{2}} {(x y - 2500)}^{2}$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $\frac{x (x y - x^{2} y' - 5000)}{{(x y - 2500)}^{2}} {(x y - 2500)}^{2}$ .

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Step 5.2.1.1

Cancel the common factor of ${(x y - 2500)}^{2}$ .

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Step 5.2.1.1.1

Cancel the common factor.

$2 y y' {(x y - 2500)}^{2} = \frac{x (x y - x^{2} y' - 5000)}{{(x y - 2500)}^{2}} {(x y - 2500)}^{2}$

Step 5.2.1.1.2

Rewrite the expression.

$2 y y' {(x y - 2500)}^{2} = x (x y - x^{2} y' - 5000)$

Step 5.2.1.2

Apply the distributive property.

$2 y y' {(x y - 2500)}^{2} = x (x y) + x (- x^{2} y') + x \cdot - 5000$

Step 5.2.1.3

Simplify.

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Step 5.2.1.3.1

Multiply $x$ by $x$ by adding the exponents.

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Step 5.2.1.3.1.1

Move $x$ .

$2 y y' {(x y - 2500)}^{2} = x \cdot x y + x (- x^{2} y') + x \cdot - 5000$

Step 5.2.1.3.1.2

Multiply $x$ by $x$ .

$2 y y' {(x y - 2500)}^{2} = x^{2} y + x (- x^{2} y') + x \cdot - 5000$

Step 5.2.1.3.2

Rewrite using the commutative property of multiplication.

$2 y y' {(x y - 2500)}^{2} = x^{2} y - x (x^{2} y') + x \cdot - 5000$

Step 5.2.1.3.3

Move $- 5000$ to the left of $x$ .

$2 y y' {(x y - 2500)}^{2} = x^{2} y - x (x^{2} y') - 5000 \cdot x$

Step 5.2.1.4

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

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Step 5.2.1.4.1

Move $x^{2}$ .

$2 y y' {(x y - 2500)}^{2} = x^{2} y - (x^{2} x) y' - 5000 \cdot x$

Step 5.2.1.4.2

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

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Step 5.2.1.4.2.1

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

$2 y y' {(x y - 2500)}^{2} = x^{2} y - (x^{2} x^{1}) y' - 5000 \cdot x$

Step 5.2.1.4.2.2

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

$2 y y' {(x y - 2500)}^{2} = x^{2} y - x^{2 + 1} y' - 5000 \cdot x$

Step 5.2.1.4.3

Add $2$ and $1$ .

$2 y y' {(x y - 2500)}^{2} = x^{2} y - x^{3} y' - 5000 \cdot x$

$2 y y' {(x y - 2500)}^{2} = x^{2} y - x^{3} y' - 5000 x$

Step 5.2.1.5

Simplify the expression.

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Step 5.2.1.5.1

Move $x^{3}$ .

$2 y y' {(x y - 2500)}^{2} = x^{2} y - y' x^{3} - 5000 x$

Step 5.2.1.5.2

Reorder $x^{2} y$ and $- y' x^{3}$ .

$2 y y' {(x y - 2500)}^{2} = - y' x^{3} + x^{2} y - 5000 x$

Step 5.3

Solve for $y'$ .

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Step 5.3.1

Move all terms containing $y'$ to the left side of the equation.

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Step 5.3.1.1

Add $y' x^{3}$ to both sides of the equation.

$2 y y' {(x y - 2500)}^{2} + y' x^{3} = x^{2} y - 5000 x$

Step 5.3.1.2

Simplify each term.

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Step 5.3.1.2.1

Rewrite ${(x y - 2500)}^{2}$ as $(x y - 2500) (x y - 2500)$ .

$2 y y' ((x y - 2500) (x y - 2500)) + y' x^{3} = x^{2} y - 5000 x$

Step 5.3.1.2.2

Expand $(x y - 2500) (x y - 2500)$ using the FOIL Method.

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Step 5.3.1.2.2.1

Apply the distributive property.

$2 y y' (x y (x y - 2500) - 2500 (x y - 2500)) + y' x^{3} = x^{2} y - 5000 x$

Step 5.3.1.2.2.2

Apply the distributive property.

$2 y y' (x y (x y) + x y \cdot - 2500 - 2500 (x y - 2500)) + y' x^{3} = x^{2} y - 5000 x$

Step 5.3.1.2.2.3

Apply the distributive property.

$2 y y' (x y (x y) + x y \cdot - 2500 - 2500 (x y) - 2500 \cdot - 2500) + y' x^{3} = x^{2} y - 5000 x$

Step 5.3.1.2.3

Simplify and combine like terms.

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Step 5.3.1.2.3.1

Simplify each term.

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Step 5.3.1.2.3.1.1

Multiply $x$ by $x$ by adding the exponents.

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Step 5.3.1.2.3.1.1.1

Move $x$ .

$2 y y' (x \cdot x y \cdot y + x y \cdot - 2500 - 2500 (x y) - 2500 \cdot - 2500) + y' x^{3} = x^{2} y - 5000 x$

Step 5.3.1.2.3.1.1.2

Multiply $x$ by $x$ .

$2 y y' (x^{2} y \cdot y + x y \cdot - 2500 - 2500 (x y) - 2500 \cdot - 2500) + y' x^{3} = x^{2} y - 5000 x$

Step 5.3.1.2.3.1.2

Multiply $y$ by $y$ by adding the exponents.

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Step 5.3.1.2.3.1.2.1

Move $y$ .

$2 y y' (x^{2} (y \cdot y) + x y \cdot - 2500 - 2500 (x y) - 2500 \cdot - 2500) + y' x^{3} = x^{2} y - 5000 x$

Step 5.3.1.2.3.1.2.2

Multiply $y$ by $y$ .

$2 y y' (x^{2} y^{2} + x y \cdot - 2500 - 2500 (x y) - 2500 \cdot - 2500) + y' x^{3} = x^{2} y - 5000 x$

Step 5.3.1.2.3.1.3

Move $- 2500$ to the left of $x y$ .

$2 y y' (x^{2} y^{2} - 2500 \cdot (x y) - 2500 (x y) - 2500 \cdot - 2500) + y' x^{3} = x^{2} y - 5000 x$

Step 5.3.1.2.3.1.4

Multiply $- 2500$ by $- 2500$ .

$2 y y' (x^{2} y^{2} - 2500 x y - 2500 x y + 6250000) + y' x^{3} = x^{2} y - 5000 x$

Step 5.3.1.2.3.2

Subtract $2500 x y$ from $- 2500 x y$ .

$2 y y' (x^{2} y^{2} - 5000 x y + 6250000) + y' x^{3} = x^{2} y - 5000 x$

Step 5.3.1.2.4

Apply the distributive property.

$2 y y' (x^{2} y^{2}) + 2 y y' (- 5000 x y) + 2 y y' \cdot 6250000 + y' x^{3} = x^{2} y - 5000 x$

Step 5.3.1.2.5

Simplify.

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Step 5.3.1.2.5.1

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

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Step 5.3.1.2.5.1.1

Move $y^{2}$ .

$2 (y^{2} y) y' x^{2} + 2 y y' (- 5000 x y) + 2 y y' \cdot 6250000 + y' x^{3} = x^{2} y - 5000 x$

Step 5.3.1.2.5.1.2

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

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Step 5.3.1.2.5.1.2.1

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

$2 (y^{2} y^{1}) y' x^{2} + 2 y y' (- 5000 x y) + 2 y y' \cdot 6250000 + y' x^{3} = x^{2} y - 5000 x$

Step 5.3.1.2.5.1.2.2

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

$2 y^{2 + 1} y' x^{2} + 2 y y' (- 5000 x y) + 2 y y' \cdot 6250000 + y' x^{3} = x^{2} y - 5000 x$

Step 5.3.1.2.5.1.3

Add $2$ and $1$ .

$2 y^{3} y' x^{2} + 2 y y' (- 5000 x y) + 2 y y' \cdot 6250000 + y' x^{3} = x^{2} y - 5000 x$

Step 5.3.1.2.5.2

Multiply $y$ by $y$ by adding the exponents.

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Step 5.3.1.2.5.2.1

Move $y$ .

$2 y^{3} y' x^{2} + 2 (y \cdot y) y' (- 5000 x) + 2 y y' \cdot 6250000 + y' x^{3} = x^{2} y - 5000 x$

Step 5.3.1.2.5.2.2

Multiply $y$ by $y$ .

$2 y^{3} y' x^{2} + 2 y^{2} y' (- 5000 x) + 2 y y' \cdot 6250000 + y' x^{3} = x^{2} y - 5000 x$

Step 5.3.1.2.5.3

Multiply $6250000$ by $2$ .

$2 y^{3} y' x^{2} + 2 y^{2} y' (- 5000 x) + 12500000 y y' + y' x^{3} = x^{2} y - 5000 x$

Step 5.3.1.2.6

Multiply $- 5000$ by $2$ .

$2 y^{3} y' x^{2} - 10000 y^{2} y' x + 12500000 y y' + y' x^{3} = x^{2} y - 5000 x$

Step 5.3.2

Factor $y'$ out of $2 y^{3} y' x^{2} - 10000 y^{2} y' x + 12500000 y y' + y' x^{3}$ .

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Step 5.3.2.1

Factor $y'$ out of $2 y^{3} y' x^{2}$ .

$y' (2 y^{3} x^{2}) - 10000 y^{2} y' x + 12500000 y y' + y' (x^{3}) = x^{2} y - 5000 x$

Step 5.3.2.2

Factor $y'$ out of $- 10000 y^{2} y' x$ .

$y' (2 y^{3} x^{2}) + y' (- 10000 y^{2} x) + 12500000 y y' + y' (x^{3}) = x^{2} y - 5000 x$

Step 5.3.2.3

Factor $y'$ out of $12500000 y y'$ .

$y' (2 y^{3} x^{2}) + y' (- 10000 y^{2} x) + y' (12500000 y) + y' (x^{3}) = x^{2} y - 5000 x$

Step 5.3.2.4

Factor $y'$ out of $y' x^{3}$ .

$y' (2 y^{3} x^{2}) + y' (- 10000 y^{2} x) + y' (12500000 y) + y' (x^{3}) = x^{2} y - 5000 x$

Step 5.3.2.5

Factor $y'$ out of $y' (2 y^{3} x^{2}) + y' (- 10000 y^{2} x)$ .

$y' (2 y^{3} x^{2} - 10000 y^{2} x) + y' (12500000 y) + y' (x^{3}) = x^{2} y - 5000 x$

Step 5.3.2.6

Factor $y'$ out of $y' (2 y^{3} x^{2} - 10000 y^{2} x) + y' (12500000 y)$ .

$y' (2 y^{3} x^{2} - 10000 y^{2} x + 12500000 y) + y' (x^{3}) = x^{2} y - 5000 x$

Step 5.3.2.7

Factor $y'$ out of $y' (2 y^{3} x^{2} - 10000 y^{2} x + 12500000 y) + y' (x^{3})$ .

$y' (2 y^{3} x^{2} - 10000 y^{2} x + 12500000 y + x^{3}) = x^{2} y - 5000 x$

Step 5.3.3

Divide each term in $y' (2 y^{3} x^{2} - 10000 y^{2} x + 12500000 y + x^{3}) = x^{2} y - 5000 x$ by $2 y^{3} x^{2} - 10000 y^{2} x + 12500000 y + x^{3}$ and simplify.

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Step 5.3.3.1

Divide each term in $y' (2 y^{3} x^{2} - 10000 y^{2} x + 12500000 y + x^{3}) = x^{2} y - 5000 x$ by $2 y^{3} x^{2} - 10000 y^{2} x + 12500000 y + x^{3}$ .

$\frac{y' (2 y^{3} x^{2} - 10000 y^{2} x + 12500000 y + x^{3})}{2 y^{3} x^{2} - 10000 y^{2} x + 12500000 y + x^{3}} = \frac{x^{2} y}{2 y^{3} x^{2} - 10000 y^{2} x + 12500000 y + x^{3}} + \frac{- 5000 x}{2 y^{3} x^{2} - 10000 y^{2} x + 12500000 y + x^{3}}$

Step 5.3.3.2

Simplify the left side.

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Step 5.3.3.2.1

Cancel the common factor of $2 y^{3} x^{2} - 10000 y^{2} x + 12500000 y + x^{3}$ .

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Step 5.3.3.2.1.1

Cancel the common factor.

Step 5.3.3.2.1.2

Divide $y'$ by $1$ .

$y' = \frac{x^{2} y}{2 y^{3} x^{2} - 10000 y^{2} x + 12500000 y + x^{3}} + \frac{- 5000 x}{2 y^{3} x^{2} - 10000 y^{2} x + 12500000 y + x^{3}}$

Step 5.3.3.3

Simplify the right side.

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Step 5.3.3.3.1

Combine the numerators over the common denominator.

$y' = \frac{x^{2} y - 5000 x}{2 y^{3} x^{2} - 10000 y^{2} x + 12500000 y + x^{3}}$

Step 5.3.3.3.2

Factor $x$ out of $x^{2} y - 5000 x$ .

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Step 5.3.3.3.2.1

Factor $x$ out of $x^{2} y$ .

$y' = \frac{x (x y) - 5000 x}{2 y^{3} x^{2} - 10000 y^{2} x + 12500000 y + x^{3}}$

Step 5.3.3.3.2.2

Factor $x$ out of $- 5000 x$ .

$y' = \frac{x (x y) + x \cdot - 5000}{2 y^{3} x^{2} - 10000 y^{2} x + 12500000 y + x^{3}}$

Step 5.3.3.3.2.3

Factor $x$ out of $x (x y) + x \cdot - 5000$ .

$y' = \frac{x (x y - 5000)}{2 y^{3} x^{2} - 10000 y^{2} x + 12500000 y + x^{3}}$

Step 6

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

$\frac{d y}{d x} = \frac{x (x y - 5000)}{2 y^{3} x^{2} - 10000 y^{2} x + 12500000 y + x^{3}}$