Calculus Examples

$3 (x^{2} + y^{2}) = 100 (x^{2} - y^{2})$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (3 (x^{2} + y^{2})) = \frac{d}{d x} (100 (x^{2} - y^{2}))$

Step 2

Differentiate the left side of the equation.

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Step 2.1

Differentiate.

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Step 2.1.1

Since $3$ is constant with respect to $x$ , the derivative of $3 (x^{2} + y^{2})$ with respect to $x$ is $3 \frac{d}{d x} [x^{2} + y^{2}]$ .

$3 \frac{d}{d x} [x^{2} + y^{2}]$

Step 2.1.2

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

$3 (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [y^{2}])$

Step 2.1.3

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

$3 (2 x + \frac{d}{d x} [y^{2}])$

Step 2.2

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 2.2.1

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

$3 (2 x + \frac{d}{d u} [u^{2}] \frac{d}{d x} [y])$

Step 2.2.2

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

$3 (2 x + 2 u \frac{d}{d x} [y])$

Step 2.2.3

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

$3 (2 x + 2 y \frac{d}{d x} [y])$

Step 2.3

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

$3 (2 x + 2 y y')$

Step 2.4

Simplify.

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Step 2.4.1

Apply the distributive property.

$3 (2 x) + 3 (2 y y')$

Step 2.4.2

Combine terms.

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Step 2.4.2.1

Multiply $2$ by $3$ .

$6 x + 3 (2 y y')$

Step 2.4.2.2

Multiply $2$ by $3$ .

$6 x + 6 y y'$

Step 2.4.3

Reorder terms.

$6 y y' + 6 x$

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

Since $100$ is constant with respect to $x$ , the derivative of $100 (x^{2} - y^{2})$ with respect to $x$ is $100 \frac{d}{d x} [x^{2} - y^{2}]$ .

$100 \frac{d}{d x} [x^{2} - y^{2}]$

Step 3.1.2

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

$100 (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- y^{2}])$

Step 3.1.3

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

$100 (2 x + \frac{d}{d x} [- y^{2}])$

Step 3.1.4

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

$100 (2 x - \frac{d}{d x} [y^{2}])$

Step 3.2

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 3.2.1

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

$100 (2 x - (\frac{d}{d u} [u^{2}] \frac{d}{d x} [y]))$

Step 3.2.2

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

$100 (2 x - (2 u \frac{d}{d x} [y]))$

Step 3.2.3

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

$100 (2 x - (2 y \frac{d}{d x} [y]))$

Step 3.3

Multiply $2$ by $- 1$ .

$100 (2 x - 2 (y \frac{d}{d x} [y]))$

Step 3.4

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

$100 (2 x - 2 y y')$

Step 3.5

Simplify.

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Step 3.5.1

Apply the distributive property.

$100 (2 x) + 100 (- 2 y y')$

Step 3.5.2

Combine terms.

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Step 3.5.2.1

Multiply $2$ by $100$ .

$200 x + 100 (- 2 y y')$

Step 3.5.2.2

Multiply $- 2$ by $100$ .

$200 x - 200 y y'$

Step 3.5.3

Reorder terms.

$- 200 y y' + 200 x$

Step 4

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

$6 y y' + 6 x = - 200 y y' + 200 x$

Step 5

Solve for $y'$ .

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Step 5.1

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

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Step 5.1.1

Add $200 y y'$ to both sides of the equation.

$6 y y' + 6 x + 200 y y' = 200 x$

Step 5.1.2

Add $6 y y'$ and $200 y y'$ .

$206 y y' + 6 x = 200 x$

Step 5.2

Move all terms not containing $y'$ to the right side of the equation.

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Step 5.2.1

Subtract $6 x$ from both sides of the equation.

$206 y y' = 200 x - 6 x$

Step 5.2.2

Subtract $6 x$ from $200 x$ .

$206 y y' = 194 x$

Step 5.3

Divide each term in $206 y y' = 194 x$ by $206 y$ and simplify.

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Step 5.3.1

Divide each term in $206 y y' = 194 x$ by $206 y$ .

$\frac{206 y y'}{206 y} = \frac{194 x}{206 y}$

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of $206$ .

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Step 5.3.2.1.1

Cancel the common factor.

$\frac{206 y y'}{206 y} = \frac{194 x}{206 y}$

Step 5.3.2.1.2

Rewrite the expression.

$\frac{y y'}{y} = \frac{194 x}{206 y}$

Step 5.3.2.2

Cancel the common factor of $y$ .

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Step 5.3.2.2.1

Cancel the common factor.

$\frac{y y'}{y} = \frac{194 x}{206 y}$

Step 5.3.2.2.2

Divide $y'$ by $1$ .

$y' = \frac{194 x}{206 y}$

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Cancel the common factor of $194$ and $206$ .

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Step 5.3.3.1.1

Factor $2$ out of $194 x$ .

$y' = \frac{2 (97 x)}{206 y}$

Step 5.3.3.1.2

Cancel the common factors.

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Step 5.3.3.1.2.1

Factor $2$ out of $206 y$ .

$y' = \frac{2 (97 x)}{2 (103 y)}$

Step 5.3.3.1.2.2

Cancel the common factor.

$y' = \frac{2 (97 x)}{2 (103 y)}$

Step 5.3.3.1.2.3

Rewrite the expression.

$y' = \frac{97 x}{103 y}$

Step 6

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

$\frac{d y}{d x} = \frac{97 x}{103 y}$