Calculus Examples

$f (x, y) = x y + y - 16 x$

Step 1

Write $f (x, y) - x y - y + 16 x = 0$ as a function.

$f (x) = f (x, y) - x y - y + 16 x$

Step 2

Move all the expressions to the left side of the equation.

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Step 2.1

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

$f (x, y) - x y = y - 16 x$

Step 2.2

Subtract $y$ from both sides of the equation.

$f (x, y) - x y - y = - 16 x$

Step 2.3

Add $16 x$ to both sides of the equation.

$f (x, y) - x y - y + 16 x = 0$

Step 3

Find the first derivative of the function.

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Step 3.1

By the Sum Rule, the derivative of $f (x, y) - x y - y + 16 x$ with respect to $x$ is $\frac{d}{d x} [f (x, y)] + \frac{d}{d x} [- x y] + \frac{d}{d x} [- y] + \frac{d}{d x} [16 x]$ .

$\frac{d}{d x} [f (x, y)] + \frac{d}{d x} [- x y] + \frac{d}{d x} [- y] + \frac{d}{d x} [16 x]$

Step 3.2

Multiply $f$ by each element of the matrix.

$\frac{d}{d x} [(f x, f y)] + \frac{d}{d x} [- x y] + \frac{d}{d x} [- y] + \frac{d}{d x} [16 x]$

Step 3.3

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

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Step 3.3.1

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

$\frac{d}{d x} [(f x, f y)] - y \frac{d}{d x} [x] + \frac{d}{d x} [- y] + \frac{d}{d x} [16 x]$

Step 3.3.2

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

$\frac{d}{d x} [(f x, f y)] - y \cdot 1 + \frac{d}{d x} [- y] + \frac{d}{d x} [16 x]$

Step 3.3.3

Multiply $- 1$ by $1$ .

$\frac{d}{d x} [(f x, f y)] - y + \frac{d}{d x} [- y] + \frac{d}{d x} [16 x]$

Step 3.4

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

$\frac{d}{d x} [(f x, f y)] - y + 0 + \frac{d}{d x} [16 x]$

Step 3.5

Evaluate $\frac{d}{d x} [16 x]$ .

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Step 3.5.1

Since $16$ is constant with respect to $x$ , the derivative of $16 x$ with respect to $x$ is $16 \frac{d}{d x} [x]$ .

$\frac{d}{d x} [(f x, f y)] - y + 0 + 16 \frac{d}{d x} [x]$

Step 3.5.2

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

$\frac{d}{d x} [(f x, f y)] - y + 0 + 16 \cdot 1$

Step 3.5.3

Multiply $16$ by $1$ .

$\frac{d}{d x} [(f x, f y)] - y + 0 + 16$

Step 3.6

Simplify.

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Step 3.6.1

Add $\frac{d}{d x} [(f x, f y)] - y$ and $0$ .

$\frac{d}{d x} [(f x, f y)] - y + 16$

Step 3.6.2

Reorder terms.

$- y + \frac{d}{d x} [(f x, f y)] + 16$

Step 4

Find the second derivative of the function.

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Step 4.1

Differentiate.

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Step 4.1.1

By the Sum Rule, the derivative of $- y + \frac{d}{d x} ((f x, f y)) + 16$ with respect to $x$ is $\frac{d}{d x} [- y] + \frac{d}{d x} [\frac{d}{d x} ((f x, f y))] + \frac{d}{d x} [16]$ .

$f'' (x) = \frac{d}{d x} (- y) + \frac{d}{d x} (\frac{d}{d x} \cdot ((f x, f y))) + \frac{d}{d x} (16)$

Step 4.1.2

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

$f'' (x) = 0 + \frac{d}{d x} (\frac{d}{d x} \cdot ((f x, f y))) + \frac{d}{d x} (16)$

Step 4.2

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

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Step 4.2.1

Cancel the common factor of $d$ .

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Step 4.2.1.1

Cancel the common factor.

$f'' (x) = 0 + \frac{d}{d x} (\frac{d}{d x} \cdot (f x, f y)) + \frac{d}{d x} (16)$

Step 4.2.1.2

Rewrite the expression.

$f'' (x) = 0 + \frac{d}{d x} (\frac{1}{x} \cdot (f x, f y)) + \frac{d}{d x} (16)$

Step 4.2.2

Multiply $\frac{1}{x}$ by each element of the matrix.

$f'' (x) = 0 + \frac{d}{d x} ((\frac{1}{x} \cdot (f x), \frac{1}{x} \cdot (f y))) + \frac{d}{d x} (16)$

Step 4.2.3

Simplify each element in the matrix.

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Step 4.2.3.1

Cancel the common factor of $x$ .

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Step 4.2.3.1.1

Factor $x$ out of $f x$ .

$f'' (x) = 0 + \frac{d}{d x} ((\frac{1}{x} \cdot (x f), \frac{1}{x} \cdot (f y))) + \frac{d}{d x} (16)$

Step 4.2.3.1.2

Cancel the common factor.

$f'' (x) = 0 + \frac{d}{d x} ((\frac{1}{x} \cdot (x f), \frac{1}{x} \cdot (f y))) + \frac{d}{d x} (16)$

Step 4.2.3.1.3

Rewrite the expression.

$f'' (x) = 0 + \frac{d}{d x} ((f, \frac{1}{x} \cdot (f y))) + \frac{d}{d x} (16)$

Step 4.2.3.2

Multiply $\frac{1}{x} (f y)$ .

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Step 4.2.3.2.1

Combine $f$ and $\frac{1}{x}$ .

$f'' (x) = 0 + \frac{d}{d x} ((f, \frac{f}{x} \cdot y)) + \frac{d}{d x} (16)$

Step 4.2.3.2.2

Combine $\frac{f}{x}$ and $y$ .

$f'' (x) = 0 + \frac{d}{d x} ((f, \frac{f y}{x})) + \frac{d}{d x} (16)$

Step 4.3

Since $16$ is constant with respect to $x$ , the derivative of $16$ with respect to $x$ is $0$ .

$f'' (x) = 0 + \frac{d}{d x} ((f, \frac{f y}{x})) + 0$

Step 4.4

Combine terms.

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Step 4.4.1

Add $0$ and $\frac{d}{d x} [(f, \frac{f y}{x})]$ .

$f'' (x) = \frac{d}{d x} ((f, \frac{f y}{x})) + 0$

Step 4.4.2

Add $\frac{d}{d x} [(f, \frac{f y}{x})]$ and $0$ .

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

Step 5

To find the local maximum and minimum values of the function, set the derivative equal to $0$ and solve.

$- y + \frac{d}{d x} ((f x, f y)) + 16 = 0$

Step 6

Find the first derivative.

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Step 6.1

Find the first derivative.

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Step 6.1.1

By the Sum Rule, the derivative of $f (x, y) - x y - y + 16 x$ with respect to $x$ is $\frac{d}{d x} [f (x, y)] + \frac{d}{d x} [- x y] + \frac{d}{d x} [- y] + \frac{d}{d x} [16 x]$ .

$\frac{d}{d x} [f (x, y)] + \frac{d}{d x} [- x y] + \frac{d}{d x} [- y] + \frac{d}{d x} [16 x]$

Step 6.1.2

Multiply $f$ by each element of the matrix.

$\frac{d}{d x} [(f x, f y)] + \frac{d}{d x} [- x y] + \frac{d}{d x} [- y] + \frac{d}{d x} [16 x]$

Step 6.1.3

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

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Step 6.1.3.1

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

$\frac{d}{d x} [(f x, f y)] - y \frac{d}{d x} [x] + \frac{d}{d x} [- y] + \frac{d}{d x} [16 x]$

Step 6.1.3.2

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

$\frac{d}{d x} [(f x, f y)] - y \cdot 1 + \frac{d}{d x} [- y] + \frac{d}{d x} [16 x]$

Step 6.1.3.3

Multiply $- 1$ by $1$ .

$\frac{d}{d x} [(f x, f y)] - y + \frac{d}{d x} [- y] + \frac{d}{d x} [16 x]$

Step 6.1.4

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

$\frac{d}{d x} [(f x, f y)] - y + 0 + \frac{d}{d x} [16 x]$

Step 6.1.5

Evaluate $\frac{d}{d x} [16 x]$ .

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Step 6.1.5.1

Since $16$ is constant with respect to $x$ , the derivative of $16 x$ with respect to $x$ is $16 \frac{d}{d x} [x]$ .

$\frac{d}{d x} [(f x, f y)] - y + 0 + 16 \frac{d}{d x} [x]$

Step 6.1.5.2

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

$\frac{d}{d x} [(f x, f y)] - y + 0 + 16 \cdot 1$

Step 6.1.5.3

Multiply $16$ by $1$ .

$\frac{d}{d x} [(f x, f y)] - y + 0 + 16$

Step 6.1.6

Simplify.

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Step 6.1.6.1

Add $\frac{d}{d x} [(f x, f y)] - y$ and $0$ .

$\frac{d}{d x} [(f x, f y)] - y + 16$

Step 6.1.6.2

Reorder terms.

$f' (x) = - y + \frac{d}{d x} [(f x, f y)] + 16$

Step 6.2

The first derivative of $f (x)$ with respect to $x$ is $- y + \frac{d}{d x} ((f x, f y)) + 16$ .

$- y + \frac{d}{d x} ((f x, f y)) + 16$

Step 7

Set the first derivative equal to $0$ then solve the equation $- y + \frac{d}{d x} ((f x, f y)) + 16 = 0$ .

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Step 7.1

Set the first derivative equal to $0$ .

$- y + \frac{d}{d x} ((f x, f y)) + 16 = 0$

Step 7.2

Simplify each term.

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Step 7.2.1

Cancel the common factor of $d$ .

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Step 7.2.1.1

Cancel the common factor.

$- y + \frac{d}{d x} \cdot (f x, f y) + 16 = 0$

Step 7.2.1.2

Rewrite the expression.

$- y + \frac{1}{x} \cdot (f x, f y) + 16 = 0$

Step 7.2.2

Multiply $\frac{1}{x}$ by each element of the matrix.

$- y + (\frac{1}{x} (f x), \frac{1}{x} (f y)) + 16 = 0$

Step 7.2.3

Simplify each element in the matrix.

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Step 7.2.3.1

Cancel the common factor of $x$ .

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Step 7.2.3.1.1

Factor $x$ out of $f x$ .

$- y + (\frac{1}{x} (x f), \frac{1}{x} (f y)) + 16 = 0$

Step 7.2.3.1.2

Cancel the common factor.

$- y + (\frac{1}{x} (x f), \frac{1}{x} (f y)) + 16 = 0$

Step 7.2.3.1.3

Rewrite the expression.

$- y + (f, \frac{1}{x} (f y)) + 16 = 0$

Step 7.2.3.2

Multiply $\frac{1}{x} (f y)$ .

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Step 7.2.3.2.1

Combine $f$ and $\frac{1}{x}$ .

$- y + (f, \frac{f}{x} y) + 16 = 0$

Step 7.2.3.2.2

Combine $\frac{f}{x}$ and $y$ .

$- y + (f, \frac{f y}{x}) + 16 = 0$

Step 7.3

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

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Step 7.3.1

Add $y$ to both sides of the equation.

$(f, \frac{f y}{x}) + 16 = y$

Step 7.3.2

Subtract $16$ from both sides of the equation.

$(f, \frac{f y}{x}) = y - 16$

Step 8

Find the values where the derivative is undefined.

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Step 8.1

Set the denominator in $\frac{d}{d x}$ equal to $0$ to find where the expression is undefined.

$d x = 0$

Step 8.2

Divide each term in $d x = 0$ by $x$ and simplify.

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Step 8.2.1

Divide each term in $d x = 0$ by $x$ .

$\frac{d x}{x} = \frac{0}{x}$

Step 8.2.2

Simplify the left side.

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Step 8.2.2.1

Cancel the common factor of $x$ .

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Step 8.2.2.1.1

Cancel the common factor.

$\frac{d x}{x} = \frac{0}{x}$

Step 8.2.2.1.2

Divide $d$ by $1$ .

$d = \frac{0}{x}$

Step 8.2.3

Simplify the right side.

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Step 8.2.3.1

Divide $0$ by $x$ .

$d = 0$

Step 8.3

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$y = 0$

Step 9

Critical points to evaluate.

$x = 0$

Step 10

Evaluate the second derivative at $x = 0$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{d}{d (0)} ((f, \frac{f y}{0}))$

Step 11

Evaluate the second derivative.

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Step 11.1

Multiply $d$ by $0$ .

$\frac{d}{0} (f, \frac{f y}{0})$

Step 11.2

The expression contains a division by $0$ . The expression is undefined.

Undefined

Step 12

Since the first derivative test failed, there are no local extrema.

No Local Extrema

Step 13