Calculus Examples

$6 x^{5} + 7 y^{5} = 13 x y$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (6 x^{5} + 7 y^{5}) = \frac{d}{d x} (13 x y)$

Step 2

Differentiate the left side of the equation.

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Step 2.1

By the Sum Rule, the derivative of $6 x^{5} + 7 y^{5}$ with respect to $x$ is $\frac{d}{d x} [6 x^{5}] + \frac{d}{d x} [7 y^{5}]$ .

$\frac{d}{d x} [6 x^{5}] + \frac{d}{d x} [7 y^{5}]$

Step 2.2

Evaluate $\frac{d}{d x} [6 x^{5}]$ .

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Step 2.2.1

Since $6$ is constant with respect to $x$ , the derivative of $6 x^{5}$ with respect to $x$ is $6 \frac{d}{d x} [x^{5}]$ .

$6 \frac{d}{d x} [x^{5}] + \frac{d}{d x} [7 y^{5}]$

Step 2.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 5$ .

$6 (5 x^{4}) + \frac{d}{d x} [7 y^{5}]$

Step 2.2.3

Multiply $5$ by $6$ .

$30 x^{4} + \frac{d}{d x} [7 y^{5}]$

Step 2.3

Evaluate $\frac{d}{d x} [7 y^{5}]$ .

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Step 2.3.1

Since $7$ is constant with respect to $x$ , the derivative of $7 y^{5}$ with respect to $x$ is $7 \frac{d}{d x} [y^{5}]$ .

$30 x^{4} + 7 \frac{d}{d x} [y^{5}]$

Step 2.3.2

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

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Step 2.3.2.1

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

$30 x^{4} + 7 (\frac{d}{d u} [u^{5}] \frac{d}{d x} [y])$

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 $n = 5$ .

$30 x^{4} + 7 (5 u^{4} \frac{d}{d x} [y])$

Step 2.3.2.3

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

$30 x^{4} + 7 (5 y^{4} \frac{d}{d x} [y])$

Step 2.3.3

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

$30 x^{4} + 7 (5 y^{4} y')$

Step 2.3.4

Multiply $5$ by $7$ .

$30 x^{4} + 35 y^{4} y'$

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.

$30 x^{4} + 35 y^{4} y' = 13 x y' + 13 y$

Step 5

Solve for $y'$ .

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Step 5.1

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

$30 x^{4} + 35 y^{4} y' - 13 x y' = 13 y$

Step 5.2

Subtract $30 x^{4}$ from both sides of the equation.

$35 y^{4} y' - 13 x y' = 13 y - 30 x^{4}$

Step 5.3

Factor $y'$ out of $35 y^{4} y' - 13 x y'$ .

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Step 5.3.1

Factor $y'$ out of $35 y^{4} y'$ .

$y' (35 y^{4}) - 13 x y' = 13 y - 30 x^{4}$

Step 5.3.2

Factor $y'$ out of $- 13 x y'$ .

$y' (35 y^{4}) + y' (- 13 x) = 13 y - 30 x^{4}$

Step 5.3.3

Factor $y'$ out of $y' (35 y^{4}) + y' (- 13 x)$ .

$y' (35 y^{4} - 13 x) = 13 y - 30 x^{4}$

Step 5.4

Divide each term in $y' (35 y^{4} - 13 x) = 13 y - 30 x^{4}$ by $35 y^{4} - 13 x$ and simplify.

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Step 5.4.1

Divide each term in $y' (35 y^{4} - 13 x) = 13 y - 30 x^{4}$ by $35 y^{4} - 13 x$ .

$\frac{y' (35 y^{4} - 13 x)}{35 y^{4} - 13 x} = \frac{13 y}{35 y^{4} - 13 x} + \frac{- 30 x^{4}}{35 y^{4} - 13 x}$

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of $35 y^{4} - 13 x$ .

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Step 5.4.2.1.1

Cancel the common factor.

$\frac{y' (35 y^{4} - 13 x)}{35 y^{4} - 13 x} = \frac{13 y}{35 y^{4} - 13 x} + \frac{- 30 x^{4}}{35 y^{4} - 13 x}$

Step 5.4.2.1.2

Divide $y'$ by $1$ .

$y' = \frac{13 y}{35 y^{4} - 13 x} + \frac{- 30 x^{4}}{35 y^{4} - 13 x}$

Step 5.4.3

Simplify the right side.

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Step 5.4.3.1

Combine the numerators over the common denominator.

$y' = \frac{13 y - 30 x^{4}}{35 y^{4} - 13 x}$

Step 6

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

$\frac{d y}{d x} = \frac{13 y - 30 x^{4}}{35 y^{4} - 13 x}$