Calculus Examples

$x y + y - 14 x$

Step 1

Write $x y + y - 14 x$ as a function.

$f (x) = x y + y - 14 x$

Step 2

Find the first derivative of the function.

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Step 2.1

By the Sum Rule, the derivative of $x y + y - 14 x$ with respect to $x$ is $\frac{d}{d x} [x y] + \frac{d}{d x} [y] + \frac{d}{d x} [- 14 x]$ .

$\frac{d}{d x} [x y] + \frac{d}{d x} [y] + \frac{d}{d x} [- 14 x]$

Step 2.2

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

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Step 2.2.1

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

$y \frac{d}{d x} [x] + \frac{d}{d x} [y] + \frac{d}{d x} [- 14 x]$

Step 2.2.2

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

$y \cdot 1 + \frac{d}{d x} [y] + \frac{d}{d x} [- 14 x]$

Step 2.2.3

Multiply $y$ by $1$ .

$y + \frac{d}{d x} [y] + \frac{d}{d x} [- 14 x]$

Step 2.3

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

$y + 0 + \frac{d}{d x} [- 14 x]$

Step 2.4

Evaluate $\frac{d}{d x} [- 14 x]$ .

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Step 2.4.1

Since $- 14$ is constant with respect to $x$ , the derivative of $- 14 x$ with respect to $x$ is $- 14 \frac{d}{d x} [x]$ .

$y + 0 - 14 \frac{d}{d x} [x]$

Step 2.4.2

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

$y + 0 - 14 \cdot 1$

Step 2.4.3

Multiply $- 14$ by $1$ .

$y + 0 - 14$

Step 2.5

Add $y$ and $0$ .

$y - 14$

Step 3

Find the second derivative of the function.

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Step 3.1

By the Sum Rule, the derivative of $y - 14$ with respect to $x$ is $\frac{d}{d x} [y] + \frac{d}{d x} [- 14]$ .

$f'' (x) = \frac{d}{d x} (y) + \frac{d}{d x} (- 14)$

Step 3.2

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

$f'' (x) = 0 + \frac{d}{d x} (- 14)$

Step 3.3

Since $- 14$ is constant with respect to $x$ , the derivative of $- 14$ with respect to $x$ is $0$ .

$f'' (x) = 0 + 0$

Step 3.4

Add $0$ and $0$ .

$f'' (x) = 0$

Step 4

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

$y - 14 = 0$

Step 5

Find the first derivative.

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Step 5.1

Find the first derivative.

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Step 5.1.1

By the Sum Rule, the derivative of $x y + y - 14 x$ with respect to $x$ is $\frac{d}{d x} [x y] + \frac{d}{d x} [y] + \frac{d}{d x} [- 14 x]$ .

$\frac{d}{d x} [x y] + \frac{d}{d x} [y] + \frac{d}{d x} [- 14 x]$

Step 5.1.2

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

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Step 5.1.2.1

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

$y \frac{d}{d x} [x] + \frac{d}{d x} [y] + \frac{d}{d x} [- 14 x]$

Step 5.1.2.2

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

$y \cdot 1 + \frac{d}{d x} [y] + \frac{d}{d x} [- 14 x]$

Step 5.1.2.3

Multiply $y$ by $1$ .

$y + \frac{d}{d x} [y] + \frac{d}{d x} [- 14 x]$

Step 5.1.3

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

$y + 0 + \frac{d}{d x} [- 14 x]$

Step 5.1.4

Evaluate $\frac{d}{d x} [- 14 x]$ .

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Step 5.1.4.1

Since $- 14$ is constant with respect to $x$ , the derivative of $- 14 x$ with respect to $x$ is $- 14 \frac{d}{d x} [x]$ .

$y + 0 - 14 \frac{d}{d x} [x]$

Step 5.1.4.2

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

$y + 0 - 14 \cdot 1$

Step 5.1.4.3

Multiply $- 14$ by $1$ .

$y + 0 - 14$

Step 5.1.5

Add $y$ and $0$ .

$f' (x) = y - 14$

Step 5.2

The first derivative of $f (x)$ with respect to $x$ is $y - 14$ .

$y - 14$

Step 6

Set the first derivative equal to $0$ then solve the equation $y - 14 = 0$ .

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Step 6.1

Set the first derivative equal to $0$ .

$y - 14 = 0$

Step 6.2

Add $14$ to both sides of the equation.

$y = 14$

Step 7

Critical points to evaluate.

$x = 14$

Step 8

Evaluate the second derivative at $x = 14$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$0$

Step 9

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

No Local Extrema

Step 10