Calculus Examples

$y = \ln (9 x^{2} + 5 y^{2})$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\ln (9 x^{2} + 5 y^{2}))$

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \ln (x)$ and $g (x) = 9 x^{2} + 5 y^{2}$ .

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Step 3.1.1

To apply the Chain Rule, set $u_{1}$ as $9 x^{2} + 5 y^{2}$ .

$\frac{d}{d u_{1}} [\ln (u_{1})] \frac{d}{d x} [9 x^{2} + 5 y^{2}]$

Step 3.1.2

The derivative of $\ln (u_{1})$ with respect to $u_{1}$ is $\frac{1}{u_{1}}$ .

$\frac{1}{u_{1}} \frac{d}{d x} [9 x^{2} + 5 y^{2}]$

Step 3.1.3

Replace all occurrences of $u_{1}$ with $9 x^{2} + 5 y^{2}$ .

$\frac{1}{9 x^{2} + 5 y^{2}} \frac{d}{d x} [9 x^{2} + 5 y^{2}]$

Step 3.2

Differentiate.

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Step 3.2.1

By the Sum Rule, the derivative of $9 x^{2} + 5 y^{2}$ with respect to $x$ is $\frac{d}{d x} [9 x^{2}] + \frac{d}{d x} [5 y^{2}]$ .

$\frac{1}{9 x^{2} + 5 y^{2}} (\frac{d}{d x} [9 x^{2}] + \frac{d}{d x} [5 y^{2}])$

Step 3.2.2

Since $9$ is constant with respect to $x$ , the derivative of $9 x^{2}$ with respect to $x$ is $9 \frac{d}{d x} [x^{2}]$ .

$\frac{1}{9 x^{2} + 5 y^{2}} (9 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [5 y^{2}])$

Step 3.2.3

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

$\frac{1}{9 x^{2} + 5 y^{2}} (9 (2 x) + \frac{d}{d x} [5 y^{2}])$

Step 3.2.4

Multiply $2$ by $9$ .

$\frac{1}{9 x^{2} + 5 y^{2}} (18 x + \frac{d}{d x} [5 y^{2}])$

Step 3.2.5

Since $5$ is constant with respect to $x$ , the derivative of $5 y^{2}$ with respect to $x$ is $5 \frac{d}{d x} [y^{2}]$ .

$\frac{1}{9 x^{2} + 5 y^{2}} (18 x + 5 \frac{d}{d x} [y^{2}])$

Step 3.3

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 3.3.1

To apply the Chain Rule, set $u_{2}$ as $y$ .

$\frac{1}{9 x^{2} + 5 y^{2}} (18 x + 5 (\frac{d}{d u_{2}} [{u_{2}}^{2}] \frac{d}{d x} [y]))$

Step 3.3.2

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

$\frac{1}{9 x^{2} + 5 y^{2}} (18 x + 5 (2 u_{2} \frac{d}{d x} [y]))$

Step 3.3.3

Replace all occurrences of $u_{2}$ with $y$ .

$\frac{1}{9 x^{2} + 5 y^{2}} (18 x + 5 (2 y \frac{d}{d x} [y]))$

Step 3.4

Multiply $2$ by $5$ .

$\frac{1}{9 x^{2} + 5 y^{2}} (18 x + 10 (y \frac{d}{d x} [y]))$

Step 3.5

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

$\frac{1}{9 x^{2} + 5 y^{2}} (18 x + 10 y y')$

Step 3.6

Simplify.

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Step 3.6.1

Reorder the factors of $\frac{1}{9 x^{2} + 5 y^{2}} (18 x + 10 y y')$ .

$(18 x + 10 y y') \frac{1}{9 x^{2} + 5 y^{2}}$

Step 3.6.2

Multiply $18 x + 10 y y'$ by $\frac{1}{9 x^{2} + 5 y^{2}}$ .

$\frac{18 x + 10 y y'}{9 x^{2} + 5 y^{2}}$

Step 3.6.3

Factor $2$ out of $18 x + 10 y y'$ .

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Step 3.6.3.1

Factor $2$ out of $18 x$ .

$\frac{2 (9 x) + 10 y y'}{9 x^{2} + 5 y^{2}}$

Step 3.6.3.2

Factor $2$ out of $10 y y'$ .

$\frac{2 (9 x) + 2 (5 y y')}{9 x^{2} + 5 y^{2}}$

Step 3.6.3.3

Factor $2$ out of $2 (9 x) + 2 (5 y y')$ .

$\frac{2 (9 x + 5 y y')}{9 x^{2} + 5 y^{2}}$

Step 4

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

$y' = \frac{2 (9 x + 5 y y')}{9 x^{2} + 5 y^{2}}$

Step 5

Solve for $y'$ .

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Step 5.1

Multiply both sides by $9 x^{2} + 5 y^{2}$ .

$y' (9 x^{2} + 5 y^{2}) = \frac{2 (9 x + 5 y y')}{9 x^{2} + 5 y^{2}} (9 x^{2} + 5 y^{2})$

Step 5.2

Simplify.

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Step 5.2.1

Simplify the left side.

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Step 5.2.1.1

Simplify $y' (9 x^{2} + 5 y^{2})$ .

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Step 5.2.1.1.1

Apply the distributive property.

$y' (9 x^{2}) + y' (5 y^{2}) = \frac{2 (9 x + 5 y y')}{9 x^{2} + 5 y^{2}} (9 x^{2} + 5 y^{2})$

Step 5.2.1.1.2

Reorder.

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Step 5.2.1.1.2.1

Rewrite using the commutative property of multiplication.

$9 y' x^{2} + y' (5 y^{2}) = \frac{2 (9 x + 5 y y')}{9 x^{2} + 5 y^{2}} (9 x^{2} + 5 y^{2})$

Step 5.2.1.1.2.2

Rewrite using the commutative property of multiplication.

$9 y' x^{2} + 5 y' y^{2} = \frac{2 (9 x + 5 y y')}{9 x^{2} + 5 y^{2}} (9 x^{2} + 5 y^{2})$

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

Simplify $\frac{2 (9 x + 5 y y')}{9 x^{2} + 5 y^{2}} (9 x^{2} + 5 y^{2})$ .

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Step 5.2.2.1.1

Cancel the common factor of $9 x^{2} + 5 y^{2}$ .

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Step 5.2.2.1.1.1

Cancel the common factor.

$9 y' x^{2} + 5 y' y^{2} = \frac{2 (9 x + 5 y y')}{9 x^{2} + 5 y^{2}} (9 x^{2} + 5 y^{2})$

Step 5.2.2.1.1.2

Rewrite the expression.

$9 y' x^{2} + 5 y' y^{2} = 2 (9 x + 5 y y')$

Step 5.2.2.1.2

Apply the distributive property.

$9 y' x^{2} + 5 y' y^{2} = 2 (9 x) + 2 (5 y y')$

Step 5.2.2.1.3

Simplify the expression.

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Step 5.2.2.1.3.1

Multiply $9$ by $2$ .

$9 y' x^{2} + 5 y' y^{2} = 18 x + 2 (5 y y')$

Step 5.2.2.1.3.2

Multiply $5$ by $2$ .

$9 y' x^{2} + 5 y' y^{2} = 18 x + 10 (y y')$

Step 5.2.2.1.3.3

Move $y$ .

$9 y' x^{2} + 5 y' y^{2} = 18 x + 10 y' y$

Step 5.2.2.1.3.4

Reorder $18 x$ and $10 y' y$ .

$9 y' x^{2} + 5 y' y^{2} = 10 y' y + 18 x$

Step 5.3

Solve for $y'$ .

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Step 5.3.1

Subtract $10 y' y$ from both sides of the equation.

$9 y' x^{2} + 5 y' y^{2} - 10 y' y = 18 x$

Step 5.3.2

Factor $y'$ out of $9 y' x^{2} + 5 y' y^{2} - 10 y' y$ .

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Step 5.3.2.1

Factor $y'$ out of $9 y' x^{2}$ .

$y' (9 x^{2}) + 5 y' y^{2} - 10 y' y = 18 x$

Step 5.3.2.2

Factor $y'$ out of $5 y' y^{2}$ .

$y' (9 x^{2}) + y' (5 y^{2}) - 10 y' y = 18 x$

Step 5.3.2.3

Factor $y'$ out of $- 10 y' y$ .

$y' (9 x^{2}) + y' (5 y^{2}) + y' (- 10 y) = 18 x$

Step 5.3.2.4

Factor $y'$ out of $y' (9 x^{2}) + y' (5 y^{2})$ .

$y' (9 x^{2} + 5 y^{2}) + y' (- 10 y) = 18 x$

Step 5.3.2.5

Factor $y'$ out of $y' (9 x^{2} + 5 y^{2}) + y' (- 10 y)$ .

$y' (9 x^{2} + 5 y^{2} - 10 y) = 18 x$

Step 5.3.3

Divide each term in $y' (9 x^{2} + 5 y^{2} - 10 y) = 18 x$ by $9 x^{2} + 5 y^{2} - 10 y$ and simplify.

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Step 5.3.3.1

Divide each term in $y' (9 x^{2} + 5 y^{2} - 10 y) = 18 x$ by $9 x^{2} + 5 y^{2} - 10 y$ .

$\frac{y' (9 x^{2} + 5 y^{2} - 10 y)}{9 x^{2} + 5 y^{2} - 10 y} = \frac{18 x}{9 x^{2} + 5 y^{2} - 10 y}$

Step 5.3.3.2

Simplify the left side.

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Step 5.3.3.2.1

Cancel the common factor of $9 x^{2} + 5 y^{2} - 10 y$ .

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Step 5.3.3.2.1.1

Cancel the common factor.

$\frac{y' (9 x^{2} + 5 y^{2} - 10 y)}{9 x^{2} + 5 y^{2} - 10 y} = \frac{18 x}{9 x^{2} + 5 y^{2} - 10 y}$

Step 5.3.3.2.1.2

Divide $y'$ by $1$ .

$y' = \frac{18 x}{9 x^{2} + 5 y^{2} - 10 y}$

Step 6

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

$\frac{d y}{d x} = \frac{18 x}{9 x^{2} + 5 y^{2} - 10 y}$