Calculus Examples

$x^{7} y + y^{7} x = 4$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (x^{7} y + y^{7} x) = \frac{d}{d x} (4)$

Step 2

Differentiate the left side of the equation.

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

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 $f (x) = x^{7}$ and $g (x) = y$ .

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

Step 2.2.2

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

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

Step 2.2.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 7$ .

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

Step 2.2.4

Move $7$ to the left of $y$ .

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

Step 2.3

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

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Step 2.3.1

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 $f (x) = y^{7}$ and $g (x) = x$ .

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

Step 2.3.2

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

$x^{7} y' + 7 y x^{6} + y^{7} \cdot 1 + x \frac{d}{d x} [y^{7}]$

Step 2.3.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{7}$ and $g (x) = y$ .

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Step 2.3.3.1

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

$x^{7} y' + 7 y x^{6} + y^{7} \cdot 1 + x (\frac{d}{d u} [u^{7}] \frac{d}{d x} [y])$

Step 2.3.3.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 7$ .

$x^{7} y' + 7 y x^{6} + y^{7} \cdot 1 + x (7 u^{6} \frac{d}{d x} [y])$

Step 2.3.3.3

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

$x^{7} y' + 7 y x^{6} + y^{7} \cdot 1 + x (7 y^{6} \frac{d}{d x} [y])$

Step 2.3.4

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

$x^{7} y' + 7 y x^{6} + y^{7} \cdot 1 + x (7 y^{6} y')$

Step 2.3.5

Multiply $y^{7}$ by $1$ .

$x^{7} y' + 7 y x^{6} + y^{7} + x (7 y^{6} y')$

Step 2.3.6

Move $7$ to the left of $x$ .

$x^{7} y' + 7 y x^{6} + y^{7} + 7 x y^{6} y'$

Step 2.4

Reorder terms.

$y^{7} + x^{7} y' + 7 x^{6} y + 7 y^{6} x y'$

Step 3

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$0$

Step 4

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

$y^{7} + x^{7} y' + 7 x^{6} y + 7 y^{6} x y' = 0$

Step 5

Solve for $y'$ .

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Step 5.1

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

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Step 5.1.1

Subtract $y^{7}$ from both sides of the equation.

$x^{7} y' + 7 x^{6} y + 7 y^{6} x y' = - y^{7}$

Step 5.1.2

Subtract $7 x^{6} y$ from both sides of the equation.

$x^{7} y' + 7 y^{6} x y' = - y^{7} - 7 x^{6} y$

Step 5.2

Factor $x y'$ out of $x^{7} y' + 7 y^{6} x y'$ .

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Step 5.2.1

Factor $x y'$ out of $x^{7} y'$ .

$x y' x^{6} + 7 y^{6} x y' = - y^{7} - 7 x^{6} y$

Step 5.2.2

Factor $x y'$ out of $7 y^{6} x y'$ .

$x y' x^{6} + x y' (7 y^{6}) = - y^{7} - 7 x^{6} y$

Step 5.2.3

Factor $x y'$ out of $x y' x^{6} + x y' (7 y^{6})$ .

$x y' (x^{6} + 7 y^{6}) = - y^{7} - 7 x^{6} y$

Step 5.3

Divide each term in $x y' (x^{6} + 7 y^{6}) = - y^{7} - 7 x^{6} y$ by $x (x^{6} + 7 y^{6})$ and simplify.

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Step 5.3.1

Divide each term in $x y' (x^{6} + 7 y^{6}) = - y^{7} - 7 x^{6} y$ by $x (x^{6} + 7 y^{6})$ .

$\frac{x y' (x^{6} + 7 y^{6})}{x (x^{6} + 7 y^{6})} = \frac{- y^{7}}{x (x^{6} + 7 y^{6})} + \frac{- 7 x^{6} y}{x (x^{6} + 7 y^{6})}$

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of $x$ .

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Step 5.3.2.1.1

Cancel the common factor.

$\frac{x y' (x^{6} + 7 y^{6})}{x (x^{6} + 7 y^{6})} = \frac{- y^{7}}{x (x^{6} + 7 y^{6})} + \frac{- 7 x^{6} y}{x (x^{6} + 7 y^{6})}$

Step 5.3.2.1.2

Rewrite the expression.

$\frac{y' (x^{6} + 7 y^{6})}{x^{6} + 7 y^{6}} = \frac{- y^{7}}{x (x^{6} + 7 y^{6})} + \frac{- 7 x^{6} y}{x (x^{6} + 7 y^{6})}$

Step 5.3.2.2

Cancel the common factor of $x^{6} + 7 y^{6}$ .

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Step 5.3.2.2.1

Cancel the common factor.

$\frac{y' (x^{6} + 7 y^{6})}{x^{6} + 7 y^{6}} = \frac{- y^{7}}{x (x^{6} + 7 y^{6})} + \frac{- 7 x^{6} y}{x (x^{6} + 7 y^{6})}$

Step 5.3.2.2.2

Divide $y'$ by $1$ .

$y' = \frac{- y^{7}}{x (x^{6} + 7 y^{6})} + \frac{- 7 x^{6} y}{x (x^{6} + 7 y^{6})}$

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Simplify each term.

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Step 5.3.3.1.1

Move the negative in front of the fraction.

$y' = - \frac{y^{7}}{x (x^{6} + 7 y^{6})} + \frac{- 7 x^{6} y}{x (x^{6} + 7 y^{6})}$

Step 5.3.3.1.2

Cancel the common factor of $x^{6}$ and $x$ .

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Step 5.3.3.1.2.1

Factor $x$ out of $- 7 x^{6} y$ .

$y' = - \frac{y^{7}}{x (x^{6} + 7 y^{6})} + \frac{x (- 7 x^{5} y)}{x (x^{6} + 7 y^{6})}$

Step 5.3.3.1.2.2

Cancel the common factors.

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Step 5.3.3.1.2.2.1

Cancel the common factor.

$y' = - \frac{y^{7}}{x (x^{6} + 7 y^{6})} + \frac{x (- 7 x^{5} y)}{x (x^{6} + 7 y^{6})}$

Step 5.3.3.1.2.2.2

Rewrite the expression.

$y' = - \frac{y^{7}}{x (x^{6} + 7 y^{6})} + \frac{- 7 x^{5} y}{x^{6} + 7 y^{6}}$

Step 5.3.3.1.3

Move the negative in front of the fraction.

$y' = - \frac{y^{7}}{x (x^{6} + 7 y^{6})} - \frac{7 x^{5} y}{x^{6} + 7 y^{6}}$

Step 5.3.3.2

To write $- \frac{7 x^{5} y}{x^{6} + 7 y^{6}}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$y' = - \frac{y^{7}}{x (x^{6} + 7 y^{6})} - \frac{7 x^{5} y}{x^{6} + 7 y^{6}} \cdot \frac{x}{x}$

Step 5.3.3.3

Write each expression with a common denominator of $x (x^{6} + 7 y^{6})$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.3.3.3.1

Multiply $\frac{7 x^{5} y}{x^{6} + 7 y^{6}}$ by $\frac{x}{x}$ .

$y' = - \frac{y^{7}}{x (x^{6} + 7 y^{6})} - \frac{7 x^{5} y x}{(x^{6} + 7 y^{6}) x}$

Step 5.3.3.3.2

Reorder the factors of $(x^{6} + 7 y^{6}) x$ .

$y' = - \frac{y^{7}}{x (x^{6} + 7 y^{6})} - \frac{7 x^{5} y x}{x (x^{6} + 7 y^{6})}$

Step 5.3.3.4

Combine the numerators over the common denominator.

$y' = \frac{- y^{7} - 7 x^{5} y x}{x (x^{6} + 7 y^{6})}$

Step 5.3.3.5

Simplify the numerator.

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Step 5.3.3.5.1

Factor $y$ out of $- y^{7} - 7 x^{5} y x$ .

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Step 5.3.3.5.1.1

Factor $y$ out of $- y^{7}$ .

$y' = \frac{y (- y^{6}) - 7 x^{5} y x}{x (x^{6} + 7 y^{6})}$

Step 5.3.3.5.1.2

Factor $y$ out of $- 7 x^{5} y x$ .

$y' = \frac{y (- y^{6}) + y (- 7 x^{5} x)}{x (x^{6} + 7 y^{6})}$

Step 5.3.3.5.1.3

Factor $y$ out of $y (- y^{6}) + y (- 7 x^{5} x)$ .

$y' = \frac{y (- y^{6} - 7 x^{5} x)}{x (x^{6} + 7 y^{6})}$

Step 5.3.3.5.2

Multiply $x^{5}$ by $x$ by adding the exponents.

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Step 5.3.3.5.2.1

Move $x$ .

$y' = \frac{y (- y^{6} - 7 (x \cdot x^{5}))}{x (x^{6} + 7 y^{6})}$

Step 5.3.3.5.2.2

Multiply $x$ by $x^{5}$ .

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Step 5.3.3.5.2.2.1

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

$y' = \frac{y (- y^{6} - 7 (x^{1} x^{5}))}{x (x^{6} + 7 y^{6})}$

Step 5.3.3.5.2.2.2

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

$y' = \frac{y (- y^{6} - 7 x^{1 + 5})}{x (x^{6} + 7 y^{6})}$

Step 5.3.3.5.2.3

Add $1$ and $5$ .

$y' = \frac{y (- y^{6} - 7 x^{6})}{x (x^{6} + 7 y^{6})}$

Step 5.3.3.6

Simplify with factoring out.

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Step 5.3.3.6.1

Factor $- 1$ out of $- y^{6}$ .

$y' = \frac{y (- (y^{6}) - 7 x^{6})}{x (x^{6} + 7 y^{6})}$

Step 5.3.3.6.2

Factor $- 1$ out of $- 7 x^{6}$ .

$y' = \frac{y (- (y^{6}) - (7 x^{6}))}{x (x^{6} + 7 y^{6})}$

Step 5.3.3.6.3

Factor $- 1$ out of $- (y^{6}) - (7 x^{6})$ .

$y' = \frac{y (- (y^{6} + 7 x^{6}))}{x (x^{6} + 7 y^{6})}$

Step 5.3.3.6.4

Simplify the expression.

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Step 5.3.3.6.4.1

Rewrite $- (y^{6} + 7 x^{6})$ as $- 1 (y^{6} + 7 x^{6})$ .

$y' = \frac{y (- 1 (y^{6} + 7 x^{6}))}{x (x^{6} + 7 y^{6})}$

Step 5.3.3.6.4.2

Move the negative in front of the fraction.

$y' = - \frac{y (y^{6} + 7 x^{6})}{x (x^{6} + 7 y^{6})}$

Step 6

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

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