Calculus Examples

$u (y) = 20 \cdot (x) + 70 \cdot (y) + \frac{{(x)}^{2}}{1000} + \frac{(x) \cdot {(y)}^{2}}{100}$

Step 1

Find the first derivative of the function.

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Step 1.1

By the Sum Rule, the derivative of $20 \cdot (x) + 70 \cdot (y) + \frac{x^{2}}{1000} + \frac{x y^{2}}{100}$ with respect to $y$ is $\frac{d}{d y} [20 \cdot (x)] + \frac{d}{d y} [70 \cdot (y)] + \frac{d}{d y} [\frac{x^{2}}{1000}] + \frac{d}{d y} [\frac{x y^{2}}{100}]$ .

$\frac{d}{d y} [20 \cdot (x)] + \frac{d}{d y} [70 \cdot (y)] + \frac{d}{d y} [\frac{x^{2}}{1000}] + \frac{d}{d y} [\frac{x y^{2}}{100}]$

Step 1.2

Since $20 x$ is constant with respect to $y$ , the derivative of $20 x$ with respect to $y$ is $0$ .

$0 + \frac{d}{d y} [70 \cdot (y)] + \frac{d}{d y} [\frac{x^{2}}{1000}] + \frac{d}{d y} [\frac{x y^{2}}{100}]$

Step 1.3

Evaluate $\frac{d}{d y} [70 \cdot (y)]$ .

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Step 1.3.1

Since $70$ is constant with respect to $y$ , the derivative of $70 y$ with respect to $y$ is $70 \frac{d}{d y} [y]$ .

$0 + 70 \frac{d}{d y} [y] + \frac{d}{d y} [\frac{x^{2}}{1000}] + \frac{d}{d y} [\frac{x y^{2}}{100}]$

Step 1.3.2

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

$0 + 70 \cdot 1 + \frac{d}{d y} [\frac{x^{2}}{1000}] + \frac{d}{d y} [\frac{x y^{2}}{100}]$

Step 1.3.3

Multiply $70$ by $1$ .

$0 + 70 + \frac{d}{d y} [\frac{x^{2}}{1000}] + \frac{d}{d y} [\frac{x y^{2}}{100}]$

Step 1.4

Since $\frac{x^{2}}{1000}$ is constant with respect to $y$ , the derivative of $\frac{x^{2}}{1000}$ with respect to $y$ is $0$ .

$0 + 70 + 0 + \frac{d}{d y} [\frac{x y^{2}}{100}]$

Step 1.5

Evaluate $\frac{d}{d y} [\frac{x y^{2}}{100}]$ .

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Step 1.5.1

Since $\frac{x}{100}$ is constant with respect to $y$ , the derivative of $\frac{x y^{2}}{100}$ with respect to $y$ is $\frac{x}{100} \frac{d}{d y} [y^{2}]$ .

$0 + 70 + 0 + \frac{x}{100} \frac{d}{d y} [y^{2}]$

Step 1.5.2

Differentiate using the Power Rule which states that $\frac{d}{d y} [y^{n}]$ is $n y^{n - 1}$ where $n = 2$ .

$0 + 70 + 0 + \frac{x}{100} (2 y)$

Step 1.5.3

Combine $2$ and $\frac{x}{100}$ .

$0 + 70 + 0 + \frac{2 x}{100} y$

Step 1.5.4

Combine $\frac{2 x}{100}$ and $y$ .

$0 + 70 + 0 + \frac{2 x y}{100}$

Step 1.5.5

Cancel the common factor of $2$ and $100$ .

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Step 1.5.5.1

Factor $2$ out of $2 x y$ .

$0 + 70 + 0 + \frac{2 (x y)}{100}$

Step 1.5.5.2

Cancel the common factors.

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Step 1.5.5.2.1

Factor $2$ out of $100$ .

$0 + 70 + 0 + \frac{2 (x y)}{2 \cdot 50}$

Step 1.5.5.2.2

Cancel the common factor.

$0 + 70 + 0 + \frac{2 (x y)}{2 \cdot 50}$

Step 1.5.5.2.3

Rewrite the expression.

$0 + 70 + 0 + \frac{x y}{50}$

Step 1.6

Simplify.

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Step 1.6.1

Combine terms.

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Step 1.6.1.1

Add $0$ and $70$ .

$70 + 0 + \frac{x y}{50}$

Step 1.6.1.2

Add $70$ and $0$ .

$70 + \frac{x y}{50}$

Step 1.6.2

Reorder terms.

$\frac{x y}{50} + 70$

Step 2

Find the second derivative of the function.

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Step 2.1

By the Sum Rule, the derivative of $\frac{x y}{50} + 70$ with respect to $y$ is $\frac{d}{d y} [\frac{x y}{50}] + \frac{d}{d y} [70]$ .

$u'' (y) = \frac{d}{d y} (\frac{x y}{50}) + \frac{d}{d y} (70)$

Step 2.2

Evaluate $\frac{d}{d y} [\frac{x y}{50}]$ .

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Step 2.2.1

Since $\frac{x}{50}$ is constant with respect to $y$ , the derivative of $\frac{x y}{50}$ with respect to $y$ is $\frac{x}{50} \frac{d}{d y} [y]$ .

$u'' (y) = \frac{x}{50} \cdot \frac{d}{d y} (y) + \frac{d}{d y} (70)$

Step 2.2.2

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

$u'' (y) = \frac{x}{50} \cdot 1 + \frac{d}{d y} (70)$

Step 2.2.3

Multiply $\frac{x}{50}$ by $1$ .

$u'' (y) = \frac{x}{50} + \frac{d}{d y} (70)$

Step 2.3

Differentiate using the Constant Rule.

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Step 2.3.1

Since $70$ is constant with respect to $y$ , the derivative of $70$ with respect to $y$ is $0$ .

$u'' (y) = \frac{x}{50} + 0$

Step 2.3.2

Add $\frac{x}{50}$ and $0$ .

$u'' (y) = \frac{x}{50}$

Step 3

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

$\frac{x y}{50} + 70 = 0$

Step 4

Find the first derivative.

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Step 4.1

Find the first derivative.

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Step 4.1.1

$\frac{d}{d y} [20 \cdot (x)] + \frac{d}{d y} [70 \cdot (y)] + \frac{d}{d y} [\frac{x^{2}}{1000}] + \frac{d}{d y} [\frac{x y^{2}}{100}]$

Step 4.1.2

Since $20 x$ is constant with respect to $y$ , the derivative of $20 x$ with respect to $y$ is $0$ .

$0 + \frac{d}{d y} [70 \cdot (y)] + \frac{d}{d y} [\frac{x^{2}}{1000}] + \frac{d}{d y} [\frac{x y^{2}}{100}]$

Step 4.1.3

Evaluate $\frac{d}{d y} [70 \cdot (y)]$ .

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Step 4.1.3.1

Since $70$ is constant with respect to $y$ , the derivative of $70 y$ with respect to $y$ is $70 \frac{d}{d y} [y]$ .

$0 + 70 \frac{d}{d y} [y] + \frac{d}{d y} [\frac{x^{2}}{1000}] + \frac{d}{d y} [\frac{x y^{2}}{100}]$

Step 4.1.3.2

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

$0 + 70 \cdot 1 + \frac{d}{d y} [\frac{x^{2}}{1000}] + \frac{d}{d y} [\frac{x y^{2}}{100}]$

Step 4.1.3.3

Multiply $70$ by $1$ .

$0 + 70 + \frac{d}{d y} [\frac{x^{2}}{1000}] + \frac{d}{d y} [\frac{x y^{2}}{100}]$

Step 4.1.4

Since $\frac{x^{2}}{1000}$ is constant with respect to $y$ , the derivative of $\frac{x^{2}}{1000}$ with respect to $y$ is $0$ .

$0 + 70 + 0 + \frac{d}{d y} [\frac{x y^{2}}{100}]$

Step 4.1.5

Evaluate $\frac{d}{d y} [\frac{x y^{2}}{100}]$ .

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Step 4.1.5.1

Since $\frac{x}{100}$ is constant with respect to $y$ , the derivative of $\frac{x y^{2}}{100}$ with respect to $y$ is $\frac{x}{100} \frac{d}{d y} [y^{2}]$ .

$0 + 70 + 0 + \frac{x}{100} \frac{d}{d y} [y^{2}]$

Step 4.1.5.2

Differentiate using the Power Rule which states that $\frac{d}{d y} [y^{n}]$ is $n y^{n - 1}$ where $n = 2$ .

$0 + 70 + 0 + \frac{x}{100} (2 y)$

Step 4.1.5.3

Combine $2$ and $\frac{x}{100}$ .

$0 + 70 + 0 + \frac{2 x}{100} y$

Step 4.1.5.4

Combine $\frac{2 x}{100}$ and $y$ .

$0 + 70 + 0 + \frac{2 x y}{100}$

Step 4.1.5.5

Cancel the common factor of $2$ and $100$ .

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Step 4.1.5.5.1

Factor $2$ out of $2 x y$ .

$0 + 70 + 0 + \frac{2 (x y)}{100}$

Step 4.1.5.5.2

Cancel the common factors.

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Step 4.1.5.5.2.1

Factor $2$ out of $100$ .

$0 + 70 + 0 + \frac{2 (x y)}{2 \cdot 50}$

Step 4.1.5.5.2.2

Cancel the common factor.

$0 + 70 + 0 + \frac{2 (x y)}{2 \cdot 50}$

Step 4.1.5.5.2.3

Rewrite the expression.

$0 + 70 + 0 + \frac{x y}{50}$

Step 4.1.6

Simplify.

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Step 4.1.6.1

Combine terms.

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Step 4.1.6.1.1

Add $0$ and $70$ .

$70 + 0 + \frac{x y}{50}$

Step 4.1.6.1.2

Add $70$ and $0$ .

$70 + \frac{x y}{50}$

Step 4.1.6.2

Reorder terms.

$f' (y) = \frac{x y}{50} + 70$

Step 4.2

The first derivative of $u (y)$ with respect to $y$ is $\frac{x y}{50} + 70$ .

$\frac{x y}{50} + 70$

Step 5

Set the first derivative equal to $0$ then solve the equation $\frac{x y}{50} + 70 = 0$ .

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Step 5.1

Set the first derivative equal to $0$ .

$\frac{x y}{50} + 70 = 0$

Step 5.2

Subtract $70$ from both sides of the equation.

$\frac{x y}{50} = - 70$

Step 5.3

Multiply both sides of the equation by $50$ .

$50 \frac{x y}{50} = 50 \cdot - 70$

Step 5.4

Simplify both sides of the equation.

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Step 5.4.1

Simplify the left side.

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Step 5.4.1.1

Cancel the common factor of $50$ .

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Step 5.4.1.1.1

Cancel the common factor.

$50 \frac{x y}{50} = 50 \cdot - 70$

Step 5.4.1.1.2

Rewrite the expression.

$x y = 50 \cdot - 70$

Step 5.4.2

Simplify the right side.

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Step 5.4.2.1

Multiply $50$ by $- 70$ .

$x y = - 3500$

Step 5.5

Divide each term in $x y = - 3500$ by $x$ and simplify.

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Step 5.5.1

Divide each term in $x y = - 3500$ by $x$ .

$\frac{x y}{x} = \frac{- 3500}{x}$

Step 5.5.2

Simplify the left side.

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Step 5.5.2.1

Cancel the common factor of $x$ .

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Step 5.5.2.1.1

Cancel the common factor.

$\frac{x y}{x} = \frac{- 3500}{x}$

Step 5.5.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 3500}{x}$

Step 5.5.3

Simplify the right side.

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Step 5.5.3.1

Move the negative in front of the fraction.

$y = - \frac{3500}{x}$

Step 6

Find the values where the derivative is undefined.

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Step 6.1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 7

Critical points to evaluate.

$y = - \frac{3500}{x}$

Step 8

Evaluate the second derivative at $y = - \frac{3500}{x}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{x}{50}$

Step 9

Since there is at least one point with $0$ or undefined second derivative, apply the first derivative test.

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Step 9.1

Split $(- \infty, \infty)$ into separate intervals around the $y$ values that make the first derivative $0$ or undefined.

$(- \infty, \infty)$

Step 9.2

Substitute any number, such as $0$ , from the interval $(- \infty, \infty)$ in the first derivative $\frac{x y}{50} + 70$ to check if the result is negative or positive.

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Step 9.2.1

Replace the variable $y$ with $0$ in the expression.

$u' (0) = \frac{x (0)}{50} + 70$

Step 9.2.2

Simplify the result.

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Step 9.2.2.1

Simplify each term.

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Step 9.2.2.1.1

Cancel the common factor of $0$ and $50$ .

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Step 9.2.2.1.1.1

Factor $50$ out of $x (0)$ .

$u' (0) = \frac{50 (x \cdot (0))}{50} + 70$

Step 9.2.2.1.1.2

Cancel the common factors.

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Step 9.2.2.1.1.2.1

Factor $50$ out of $50$ .

$u' (0) = \frac{50 (x \cdot (0))}{50 (1)} + 70$

Step 9.2.2.1.1.2.2

Cancel the common factor.

$u' (0) = \frac{50 (x \cdot (0))}{50 \cdot 1} + 70$

Step 9.2.2.1.1.2.3

Rewrite the expression.

$u' (0) = \frac{x \cdot (0)}{1} + 70$

Step 9.2.2.1.1.2.4

Divide $x \cdot (0)$ by $1$ .

$u' (0) = x \cdot (0) + 70$

Step 9.2.2.1.2

Multiply $x$ by $0$ .

$u' (0) = 0 + 70$

Step 9.2.2.2

Add $0$ and $70$ .

$u' (0) = 70$

Step 9.2.2.3

The final answer is $70$ .

$70$

Step 9.3

No local maxima or minima found for $u (y) = 20 \cdot (x) + 70 \cdot (y) + \frac{{(x)}^{2}}{1000} + \frac{(x) \cdot {(y)}^{2}}{100}$ .

No local maxima or minima

Step 10