Calculus Examples

$f (x, y) = x + y + x y$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (f (x, y)) = \frac{d}{d x} (x + y + x y)$

Step 2

Multiply $f$ by each element of the matrix.

$\frac{d}{d x} [(f x, f y)]$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

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

Step 3.2

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

$1 + \frac{d}{d x} [y] + \frac{d}{d x} [x y]$

Step 3.3

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

$1 + y' + \frac{d}{d x} [x y]$

Step 3.4

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

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Step 3.4.1

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 $f (x) = x$ and $g (x) = y$ .

$1 + y' + x \frac{d}{d x} [y] + y \frac{d}{d x} [x]$

Step 3.4.2

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

$1 + y' + x y' + y \frac{d}{d x} [x]$

Step 3.4.3

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

$1 + y' + x y' + y \cdot 1$

Step 3.4.4

Multiply $y$ by $1$ .

$1 + y' + x y' + y$

Step 3.5

Reorder terms.

$x y' + y + 1 + y'$

Step 4

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

$\frac{d}{d x} ((f x, f y)) = x y' + y + 1 + y'$

Step 5

Solve for $y'$ .

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Step 5.1

Rewrite the equation as $x y' + y + 1 + y' = \frac{d}{d x ((f x, f y))}$ .

$x y' + y + 1 + y' = \frac{d}{d x ((f x, f y))}$

Step 5.2

Simplify $\frac{d}{d x ((f x, f y))}$ .

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Step 5.2.1

Cancel the common factor of $d$ .

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Step 5.2.1.1

Cancel the common factor.

$x y' + y + 1 + y' = \frac{d}{d x (f x, f y)}$

Step 5.2.1.2

Rewrite the expression.

$x y' + y + 1 + y' = \frac{1}{x ((f x, f y))}$

Step 5.2.2

Simplify the denominator.

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Step 5.2.2.1

Multiply $x$ by each element of the matrix.

$x y' + y + 1 + y' = \frac{1}{(x (f x), x (f y))}$

Step 5.2.2.2

Multiply $x$ by $x$ by adding the exponents.

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Step 5.2.2.2.1

Move $x$ .

$x y' + y + 1 + y' = \frac{1}{(x \cdot x f, x (f y))}$

Step 5.2.2.2.2

Multiply $x$ by $x$ .

$x y' + y + 1 + y' = \frac{1}{(x^{2} f, x (f y))}$

$x y' + y + 1 + y' = \frac{1}{(x^{2} f, x f y)}$

Step 5.3

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

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Step 5.3.1

Subtract $y$ from both sides of the equation.

$x y' + 1 + y' = \frac{1}{(x^{2} f, x f y)} - y$

Step 5.3.2

Subtract $1$ from both sides of the equation.

$x y' + y' = \frac{1}{(x^{2} f, x f y)} - y - 1$

Step 5.4

Factor $y'$ out of $x y' + y'$ .

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Step 5.4.1

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

$y' x + y' = \frac{1}{(x^{2} f, x f y)} - y - 1$

Step 5.4.2

Raise $y'$ to the power of $1$ .

$y' x + y' = \frac{1}{(x^{2} f, x f y)} - y - 1$

Step 5.4.3

Factor $y'$ out of ${y'}^{1}$ .

$y' x + y' \cdot 1 = \frac{1}{(x^{2} f, x f y)} - y - 1$

Step 5.4.4

Factor $y'$ out of $y' x + y' \cdot 1$ .

$y' (x + 1) = \frac{1}{(x^{2} f, x f y)} - y - 1$

Step 5.5

Divide each term in $y' (x + 1) = \frac{1}{(x^{2} f, x f y)} - y - 1$ by $x + 1$ and simplify.

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Step 5.5.1

Divide each term in $y' (x + 1) = \frac{1}{(x^{2} f, x f y)} - y - 1$ by $x + 1$ .

$\frac{y' (x + 1)}{x + 1} = \frac{\frac{1}{(x^{2} f, x f y)}}{x + 1} + \frac{- y}{x + 1} + \frac{- 1}{x + 1}$

Step 5.5.2

Simplify the left side.

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Step 5.5.2.1

Cancel the common factor of $x + 1$ .

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Step 5.5.2.1.1

Cancel the common factor.

$\frac{y' (x + 1)}{x + 1} = \frac{\frac{1}{(x^{2} f, x f y)}}{x + 1} + \frac{- y}{x + 1} + \frac{- 1}{x + 1}$

Step 5.5.2.1.2

Divide $y'$ by $1$ .

$y' = \frac{\frac{1}{(x^{2} f, x f y)}}{x + 1} + \frac{- y}{x + 1} + \frac{- 1}{x + 1}$

Step 5.5.3

Simplify the right side.

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Step 5.5.3.1

Simplify each term.

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Step 5.5.3.1.1

Multiply the numerator by the reciprocal of the denominator.

$y' = \frac{1}{(x^{2} f, x f y)} \cdot \frac{1}{x + 1} + \frac{- y}{x + 1} + \frac{- 1}{x + 1}$

Step 5.5.3.1.2

Multiply $\frac{1}{(x^{2} f, x f y)}$ by $\frac{1}{x + 1}$ .

$y' = \frac{1}{(x^{2} f, x f y) (x + 1)} + \frac{- y}{x + 1} + \frac{- 1}{x + 1}$

Step 5.5.3.1.3

Move the negative in front of the fraction.

$y' = \frac{1}{(x^{2} f, x f y) (x + 1)} - \frac{y}{x + 1} + \frac{- 1}{x + 1}$

Step 5.5.3.1.4

Move the negative in front of the fraction.

$y' = \frac{1}{(x^{2} f, x f y) (x + 1)} - \frac{y}{x + 1} - \frac{1}{x + 1}$

Step 5.5.3.2

Combine into one fraction.

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Step 5.5.3.2.1

Combine the numerators over the common denominator.

$y' = \frac{1}{(x^{2} f, x f y) (x + 1)} + \frac{- y - 1}{x + 1}$

Step 5.5.3.2.2

Reorder factors in $\frac{1}{(x^{2} f, x f y) (x + 1)} + \frac{- y - 1}{x + 1}$ .

$y' = \frac{1}{(x + 1) (x^{2} f, x f y)} + \frac{- y - 1}{x + 1}$

Step 6

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

$\frac{d y}{d x} = \frac{1}{(x + 1) (x^{2} f, x f y)} + \frac{- y - 1}{x + 1}$