Calculus Examples

6 x^{3} + 7 y^{3} = 13 x y

$6 x^{3} + 7 y^{3} = 13 x y$

Step 1

Differentiate both sides of the equation.

\frac{d}{d x} (6 x^{3} + 7 y^{3}) = \frac{d}{d x} (13 x y)

$\frac{d}{d x} (6 x^{3} + 7 y^{3}) = \frac{d}{d x} (13 x y)$

Step 2

Differentiate the left side of the equation.

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Step 2.1

By the Sum Rule, the derivative of

6 x^{3} + 7 y^{3}

$6 x^{3} + 7 y^{3}$ with respect to

x

$x$ is

\frac{d}{d x} [6 x^{3}] + \frac{d}{d x} [7 y^{3}]

$\frac{d}{d x} [6 x^{3}] + \frac{d}{d x} [7 y^{3}]$ .

\frac{d}{d x} [6 x^{3}] + \frac{d}{d x} [7 y^{3}]

$\frac{d}{d x} [6 x^{3}] + \frac{d}{d x} [7 y^{3}]$

Step 2.2

Evaluate

\frac{d}{d x} [6 x^{3}]

$\frac{d}{d x} [6 x^{3}]$ .

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Step 2.2.1

Since

6

$6$ is constant with respect to

x

$x$ , the derivative of

6 x^{3}

$6 x^{3}$ with respect to

x

$x$ is

6 \frac{d}{d x} [x^{3}]

$6 \frac{d}{d x} [x^{3}]$ .

6 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [7 y^{3}]

$6 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [7 y^{3}]$

Step 2.2.2

Differentiate using the Power Rule which states that

\frac{d}{d x} [x^{n}]

$\frac{d}{d x} [x^{n}]$ is

n x^{n - 1}

$n x^{n - 1}$ where

n = 3

$n = 3$ .

6 (3 x^{2}) + \frac{d}{d x} [7 y^{3}]

$6 (3 x^{2}) + \frac{d}{d x} [7 y^{3}]$

Step 2.2.3

Multiply

3

$3$ by

6

$6$ .

18 x^{2} + \frac{d}{d x} [7 y^{3}]

$18 x^{2} + \frac{d}{d x} [7 y^{3}]$

18 x^{2} + \frac{d}{d x} [7 y^{3}]

$18 x^{2} + \frac{d}{d x} [7 y^{3}]$

Step 2.3

Evaluate

\frac{d}{d x} [7 y^{3}]

$\frac{d}{d x} [7 y^{3}]$ .

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Step 2.3.1

Since

7

$7$ is constant with respect to

x

$x$ , the derivative of

7 y^{3}

$7 y^{3}$ with respect to

x

$x$ is

7 \frac{d}{d x} [y^{3}]

$7 \frac{d}{d x} [y^{3}]$ .

18 x^{2} + 7 \frac{d}{d x} [y^{3}]

$18 x^{2} + 7 \frac{d}{d x} [y^{3}]$

Step 2.3.2

Differentiate using the chain rule, which states that

\frac{d}{d x} [f (g (x))]

$\frac{d}{d x} [f (g (x))]$ is

$f' (g (x)) g' (x)$ where

$f (x) = x^{3}$ and

$g (x) = y$ .

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Step 2.3.2.1

To apply the Chain Rule, set

$u$ as

$y$ .

$18 x^{2} + 7 (\frac{d}{d u} [u^{3}] \frac{d}{d x} [y])$

Step 2.3.2.2

Differentiate using the Power Rule which states that

$\frac{d}{d u} [u^{n}]$ is

$n u^{n - 1}$ where

$n = 3$ .

$18 x^{2} + 7 (3 u^{2} \frac{d}{d x} [y])$

Step 2.3.2.3

Replace all occurrences of

$u$ with

$y$ .

$18 x^{2} + 7 (3 y^{2} \frac{d}{d x} [y])$

Step 2.3.3

Rewrite

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

$y'$ .

$18 x^{2} + 7 (3 y^{2} y')$

Step 2.3.4

Multiply

$3$ by

$7$ .

$18 x^{2} + 21 y^{2} y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Since

$13$ is constant with respect to

$x$ , the derivative of

$13 x y$ with respect to

$x$ is

$13 \frac{d}{d x} [x y]$ .

$13 \frac{d}{d x} [x y]$

Step 3.2

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

$g (x) = y$ .

$13 (x \frac{d}{d x} [y] + y \frac{d}{d x} [x])$

Step 3.3

Rewrite

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

$y'$ .

$13 (x y' + y \frac{d}{d x} [x])$

Step 3.4

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 1$ .

$13 (x y' + y \cdot 1)$

Step 3.5

Multiply

$y$ by

$1$ .

$13 (x y' + y)$

Step 3.6

Apply the distributive property.

$13 x y' + 13 y$

Step 4

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

$18 x^{2} + 21 y^{2} y' = 13 x y' + 13 y$

Step 5

Solve for

$y'$ .

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Step 5.1

Subtract

$13 x y'$ from both sides of the equation.

$18 x^{2} + 21 y^{2} y' - 13 x y' = 13 y$

Step 5.2

Subtract

$18 x^{2}$ from both sides of the equation.

$21 y^{2} y' - 13 x y' = 13 y - 18 x^{2}$

Step 5.3

Factor

$y'$ out of

$21 y^{2} y' - 13 x y'$ .

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Step 5.3.1

Factor

$y'$ out of

$21 y^{2} y'$ .

$y' (21 y^{2}) - 13 x y' = 13 y - 18 x^{2}$

Step 5.3.2

Factor

$y'$ out of

$- 13 x y'$ .

$y' (21 y^{2}) + y' (- 13 x) = 13 y - 18 x^{2}$

Step 5.3.3

Factor

$y'$ out of

$y' (21 y^{2}) + y' (- 13 x)$ .

$y' (21 y^{2} - 13 x) = 13 y - 18 x^{2}$

Step 5.4

Divide each term in

$y' (21 y^{2} - 13 x) = 13 y - 18 x^{2}$ by

$21 y^{2} - 13 x$ and simplify.

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Step 5.4.1

Divide each term in

$y' (21 y^{2} - 13 x) = 13 y - 18 x^{2}$ by

$21 y^{2} - 13 x$ .

$\frac{y' (21 y^{2} - 13 x)}{21 y^{2} - 13 x} = \frac{13 y}{21 y^{2} - 13 x} + \frac{- 18 x^{2}}{21 y^{2} - 13 x}$

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of

$21 y^{2} - 13 x$ .

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Step 5.4.2.1.1

Cancel the common factor.

$\frac{y' (21 y^{2} - 13 x)}{21 y^{2} - 13 x} = \frac{13 y}{21 y^{2} - 13 x} + \frac{- 18 x^{2}}{21 y^{2} - 13 x}$

Step 5.4.2.1.2

Divide

$y'$ by

$1$ .

$y' = \frac{13 y}{21 y^{2} - 13 x} + \frac{- 18 x^{2}}{21 y^{2} - 13 x}$

Step 5.4.3

Simplify the right side.

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Step 5.4.3.1

Combine the numerators over the common denominator.

$y' = \frac{13 y - 18 x^{2}}{21 y^{2} - 13 x}$

Step 6

Replace

$y'$ with

$\frac{d y}{d x}$ .

$\frac{d y}{d x} = \frac{13 y - 18 x^{2}}{21 y^{2} - 13 x}$