Calculus Examples

$f (x) = 2 x^{2} \cdot (3 x y) + 4 y^{2} - 2 x + 10 y$

Step 1

Find the first derivative of the function.

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Step 1.1

By the Sum Rule, the derivative of $2 x^{2} \cdot (3 x y) + 4 y^{2} - 2 x + 10 y$ with respect to $x$ is $\frac{d}{d x} [2 x^{2} \cdot (3 x y)] + \frac{d}{d x} [4 y^{2}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [10 y]$ .

$\frac{d}{d x} [2 x^{2} \cdot (3 x y)] + \frac{d}{d x} [4 y^{2}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [10 y]$

Step 1.2

Evaluate $\frac{d}{d x} [2 x^{2} \cdot (3 x y)]$ .

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Step 1.2.1

Multiply $3$ by $2$ .

$\frac{d}{d x} [6 x^{2} \cdot (x y)] + \frac{d}{d x} [4 y^{2}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [10 y]$

Step 1.2.2

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

$\frac{d}{d x} [6 (x^{1} x^{2}) \cdot y] + \frac{d}{d x} [4 y^{2}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [10 y]$

Step 1.2.3

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

$\frac{d}{d x} [6 x^{1 + 2} \cdot y] + \frac{d}{d x} [4 y^{2}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [10 y]$

Step 1.2.4

Add $1$ and $2$ .

$\frac{d}{d x} [6 x^{3} \cdot y] + \frac{d}{d x} [4 y^{2}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [10 y]$

Step 1.2.5

Since $6 y$ is constant with respect to $x$ , the derivative of $6 x^{3} y$ with respect to $x$ is $6 y \frac{d}{d x} [x^{3}]$ .

$6 y \frac{d}{d x} [x^{3}] + \frac{d}{d x} [4 y^{2}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [10 y]$

Step 1.2.6

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$6 y (3 x^{2}) + \frac{d}{d x} [4 y^{2}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [10 y]$

Step 1.2.7

Multiply $3$ by $6$ .

$18 y x^{2} + \frac{d}{d x} [4 y^{2}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [10 y]$

Step 1.3

Since $4 y^{2}$ is constant with respect to $x$ , the derivative of $4 y^{2}$ with respect to $x$ is $0$ .

$18 y x^{2} + 0 + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [10 y]$

Step 1.4

Evaluate $\frac{d}{d x} [- 2 x]$ .

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Step 1.4.1

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

$18 y x^{2} + 0 - 2 \frac{d}{d x} [x] + \frac{d}{d x} [10 y]$

Step 1.4.2

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

$18 y x^{2} + 0 - 2 \cdot 1 + \frac{d}{d x} [10 y]$

Step 1.4.3

Multiply $- 2$ by $1$ .

$18 y x^{2} + 0 - 2 + \frac{d}{d x} [10 y]$

Step 1.5

Since $10 y$ is constant with respect to $x$ , the derivative of $10 y$ with respect to $x$ is $0$ .

$18 y x^{2} + 0 - 2 + 0$

Step 1.6

Simplify.

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Step 1.6.1

Combine terms.

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Step 1.6.1.1

Add $18 y x^{2}$ and $0$ .

$18 y x^{2} - 2 + 0$

Step 1.6.1.2

Add $18 y x^{2} - 2$ and $0$ .

$18 y x^{2} - 2$

Step 1.6.2

Reorder terms.

$18 x^{2} y - 2$

Step 2

Find the second derivative of the function.

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Step 2.1

By the Sum Rule, the derivative of $18 x^{2} y - 2$ with respect to $x$ is $\frac{d}{d x} [18 x^{2} y] + \frac{d}{d x} [- 2]$ .

$f'' (x) = \frac{d}{d x} (18 x^{2} y) + \frac{d}{d x} (- 2)$

Step 2.2

Evaluate $\frac{d}{d x} [18 x^{2} y]$ .

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Step 2.2.1

Since $18 y$ is constant with respect to $x$ , the derivative of $18 x^{2} y$ with respect to $x$ is $18 y \frac{d}{d x} [x^{2}]$ .

$f'' (x) = 18 y \frac{d}{d x} (x^{2}) + \frac{d}{d x} (- 2)$

Step 2.2.2

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

$f'' (x) = 18 y (2 x) + \frac{d}{d x} (- 2)$

Step 2.2.3

Multiply $2$ by $18$ .

$f'' (x) = 36 y x + \frac{d}{d x} (- 2)$

Step 2.3

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

$f'' (x) = 36 y x + 0$

Step 2.4

Simplify.

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Step 2.4.1

Add $36 y x$ and $0$ .

$f'' (x) = 36 y x$

Step 2.4.2

Reorder the factors of $36 y x$ .

$f'' (x) = 36 x y$

Step 3

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

$18 x^{2} y - 2 = 0$

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

$\frac{d}{d x} [2 x^{2} \cdot (3 x y)] + \frac{d}{d x} [4 y^{2}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [10 y]$

Step 4.1.2

Evaluate $\frac{d}{d x} [2 x^{2} \cdot (3 x y)]$ .

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Step 4.1.2.1

Multiply $3$ by $2$ .

$\frac{d}{d x} [6 x^{2} \cdot (x y)] + \frac{d}{d x} [4 y^{2}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [10 y]$

Step 4.1.2.2

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

$\frac{d}{d x} [6 (x^{1} x^{2}) \cdot y] + \frac{d}{d x} [4 y^{2}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [10 y]$

Step 4.1.2.3

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

$\frac{d}{d x} [6 x^{1 + 2} \cdot y] + \frac{d}{d x} [4 y^{2}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [10 y]$

Step 4.1.2.4

Add $1$ and $2$ .

$\frac{d}{d x} [6 x^{3} \cdot y] + \frac{d}{d x} [4 y^{2}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [10 y]$

Step 4.1.2.5

Since $6 y$ is constant with respect to $x$ , the derivative of $6 x^{3} y$ with respect to $x$ is $6 y \frac{d}{d x} [x^{3}]$ .

$6 y \frac{d}{d x} [x^{3}] + \frac{d}{d x} [4 y^{2}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [10 y]$

Step 4.1.2.6

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$6 y (3 x^{2}) + \frac{d}{d x} [4 y^{2}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [10 y]$

Step 4.1.2.7

Multiply $3$ by $6$ .

$18 y x^{2} + \frac{d}{d x} [4 y^{2}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [10 y]$

Step 4.1.3

Since $4 y^{2}$ is constant with respect to $x$ , the derivative of $4 y^{2}$ with respect to $x$ is $0$ .

$18 y x^{2} + 0 + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [10 y]$

Step 4.1.4

Evaluate $\frac{d}{d x} [- 2 x]$ .

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Step 4.1.4.1

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

$18 y x^{2} + 0 - 2 \frac{d}{d x} [x] + \frac{d}{d x} [10 y]$

Step 4.1.4.2

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

$18 y x^{2} + 0 - 2 \cdot 1 + \frac{d}{d x} [10 y]$

Step 4.1.4.3

Multiply $- 2$ by $1$ .

$18 y x^{2} + 0 - 2 + \frac{d}{d x} [10 y]$

Step 4.1.5

Since $10 y$ is constant with respect to $x$ , the derivative of $10 y$ with respect to $x$ is $0$ .

$18 y x^{2} + 0 - 2 + 0$

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 $18 y x^{2}$ and $0$ .

$18 y x^{2} - 2 + 0$

Step 4.1.6.1.2

Add $18 y x^{2} - 2$ and $0$ .

$18 y x^{2} - 2$

Step 4.1.6.2

Reorder terms.

$f' (x) = 18 x^{2} y - 2$

Step 4.2

The first derivative of $f (x)$ with respect to $x$ is $18 x^{2} y - 2$ .

$18 x^{2} y - 2$

Step 5

Set the first derivative equal to $0$ then solve the equation $18 x^{2} y - 2 = 0$ .

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Step 5.1

Set the first derivative equal to $0$ .

$18 x^{2} y - 2 = 0$

Step 5.2

Add $2$ to both sides of the equation.

$18 x^{2} y = 2$

Step 5.3

Divide each term in $18 x^{2} y = 2$ by $18 y$ and simplify.

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Step 5.3.1

Divide each term in $18 x^{2} y = 2$ by $18 y$ .

$\frac{18 x^{2} y}{18 y} = \frac{2}{18 y}$

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of $18$ .

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Step 5.3.2.1.1

Cancel the common factor.

$\frac{18 x^{2} y}{18 y} = \frac{2}{18 y}$

Step 5.3.2.1.2

Rewrite the expression.

$\frac{x^{2} y}{y} = \frac{2}{18 y}$

Step 5.3.2.2

Cancel the common factor of $y$ .

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Step 5.3.2.2.1

Cancel the common factor.

$\frac{x^{2} y}{y} = \frac{2}{18 y}$

Step 5.3.2.2.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{2}{18 y}$

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Cancel the common factor of $2$ and $18$ .

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Step 5.3.3.1.1

Factor $2$ out of $2$ .

$x^{2} = \frac{2 (1)}{18 y}$

Step 5.3.3.1.2

Cancel the common factors.

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Step 5.3.3.1.2.1

Factor $2$ out of $18 y$ .

$x^{2} = \frac{2 (1)}{2 (9 y)}$

Step 5.3.3.1.2.2

Cancel the common factor.

$x^{2} = \frac{2 \cdot 1}{2 (9 y)}$

Step 5.3.3.1.2.3

Rewrite the expression.

$x^{2} = \frac{1}{9 y}$

Step 5.4

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{\frac{1}{9 y}}$

Step 5.5

Simplify $\pm \sqrt{\frac{1}{9 y}}$ .

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Step 5.5.1

Rewrite $\frac{1}{9 y}$ as ${(\frac{1}{3})}^{2} \frac{1}{y}$ .

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Step 5.5.1.1

Factor the perfect power $1^{2}$ out of $1$ .

$x = \pm \sqrt{\frac{1^{2} \cdot 1}{9 y}}$

Step 5.5.1.2

Factor the perfect power $3^{2}$ out of $9 y$ .

$x = \pm \sqrt{\frac{1^{2} \cdot 1}{3^{2} y}}$

Step 5.5.1.3

Rearrange the fraction $\frac{1^{2} \cdot 1}{3^{2} y}$ .

$x = \pm \sqrt{{(\frac{1}{3})}^{2} \frac{1}{y}}$

Step 5.5.2

Pull terms out from under the radical.

$x = \pm \frac{1}{3} \sqrt{\frac{1}{y}}$

Step 5.5.3

Rewrite $\sqrt{\frac{1}{y}}$ as $\frac{\sqrt{1}}{\sqrt{y}}$ .

$x = \pm \frac{1}{3} \cdot \frac{\sqrt{1}}{\sqrt{y}}$

Step 5.5.4

Any root of $1$ is $1$ .

$x = \pm \frac{1}{3} \cdot \frac{1}{\sqrt{y}}$

Step 5.5.5

Multiply $\frac{1}{\sqrt{y}}$ by $\frac{\sqrt{y}}{\sqrt{y}}$ .

$x = \pm \frac{1}{3} (\frac{1}{\sqrt{y}} \cdot \frac{\sqrt{y}}{\sqrt{y}})$

Step 5.5.6

Combine and simplify the denominator.

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Step 5.5.6.1

Multiply $\frac{1}{\sqrt{y}}$ by $\frac{\sqrt{y}}{\sqrt{y}}$ .

$x = \pm \frac{1}{3} \cdot \frac{\sqrt{y}}{\sqrt{y} \sqrt{y}}$

Step 5.5.6.2

Raise $\sqrt{y}$ to the power of $1$ .

$x = \pm \frac{1}{3} \cdot \frac{\sqrt{y}}{{\sqrt{y}}^{1} \sqrt{y}}$

Step 5.5.6.3

Raise $\sqrt{y}$ to the power of $1$ .

$x = \pm \frac{1}{3} \cdot \frac{\sqrt{y}}{{\sqrt{y}}^{1} {\sqrt{y}}^{1}}$

Step 5.5.6.4

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

$x = \pm \frac{1}{3} \cdot \frac{\sqrt{y}}{{\sqrt{y}}^{1 + 1}}$

Step 5.5.6.5

Add $1$ and $1$ .

$x = \pm \frac{1}{3} \cdot \frac{\sqrt{y}}{{\sqrt{y}}^{2}}$

Step 5.5.6.6

Rewrite ${\sqrt{y}}^{2}$ as $y$ .

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Step 5.5.6.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{y}$ as $y^{\frac{1}{2}}$ .

$x = \pm \frac{1}{3} \cdot \frac{\sqrt{y}}{{(y^{\frac{1}{2}})}^{2}}$

Step 5.5.6.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = \pm \frac{1}{3} \cdot \frac{\sqrt{y}}{y^{\frac{1}{2} \cdot 2}}$

Step 5.5.6.6.3

Combine $\frac{1}{2}$ and $2$ .

$x = \pm \frac{1}{3} \cdot \frac{\sqrt{y}}{y^{\frac{2}{2}}}$

Step 5.5.6.6.4

Cancel the common factor of $2$ .

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Step 5.5.6.6.4.1

Cancel the common factor.

$x = \pm \frac{1}{3} \cdot \frac{\sqrt{y}}{y^{\frac{2}{2}}}$

Step 5.5.6.6.4.2

Rewrite the expression.

$x = \pm \frac{1}{3} \cdot \frac{\sqrt{y}}{y^{1}}$

Step 5.5.6.6.5

Simplify.

$x = \pm \frac{1}{3} \cdot \frac{\sqrt{y}}{y}$

Step 5.5.7

Multiply $\frac{1}{3}$ by $\frac{\sqrt{y}}{y}$ .

$x = \pm \frac{\sqrt{y}}{3 y}$

Step 5.6

The complete solution is the result of both the positive and negative portions of the solution.

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Step 5.6.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \frac{\sqrt{y}}{3 y}$

Step 5.6.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \frac{\sqrt{y}}{3 y}$

Step 5.6.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = \frac{\sqrt{y}}{3 y}$

$x = - \frac{\sqrt{y}}{3 y}$

$x = \frac{\sqrt{y}}{3 y}$

$x = - \frac{\sqrt{y}}{3 y}$

$x = \frac{\sqrt{y}}{3 y}$

$x = - \frac{\sqrt{y}}{3 y}$

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.

$x = \frac{\sqrt{y}}{3 y}$

$x = - \frac{\sqrt{y}}{3 y}$

Step 8

Evaluate the second derivative at $x = \frac{\sqrt{y}}{3 y}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$36 (\frac{\sqrt{y}}{3 y}) y$

Step 9

Evaluate the second derivative.

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Step 9.1

Cancel the common factor of $3$ .

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Step 9.1.1

Factor $3$ out of $36$ .

$3 (12) \frac{\sqrt{y}}{3 y} y$

Step 9.1.2

Factor $3$ out of $3 y$ .

$3 (12) \frac{\sqrt{y}}{3 (y)} y$

Step 9.1.3

Cancel the common factor.

$3 \cdot 12 \frac{\sqrt{y}}{3 y} y$

Step 9.1.4

Rewrite the expression.

$12 \frac{\sqrt{y}}{y} y$

Step 9.2

Combine $12$ and $\frac{\sqrt{y}}{y}$ .

$\frac{12 \sqrt{y}}{y} y$

Step 9.3

Cancel the common factor of $y$ .

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Step 9.3.1

Cancel the common factor.

$\frac{12 \sqrt{y}}{y} y$

Step 9.3.2

Rewrite the expression.

$12 \sqrt{y}$

Step 10

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

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Step 10.1

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

$(- \infty, \infty)$

Step 10.2

Substitute any number, such as $0$ , from the interval $(- \infty, \infty)$ in the first derivative $18 x^{2} y - 2$ to check if the result is negative or positive.

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Step 10.2.1

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

$f' (0) = 18 {(0)}^{2} y - 2$

Step 10.2.2

Simplify the result.

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Step 10.2.2.1

Simplify each term.

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Step 10.2.2.1.1

Raising $0$ to any positive power yields $0$ .

$f' (0) = 18 \cdot (0 y) - 2$

Step 10.2.2.1.2

Multiply $18$ by $0$ .

$f' (0) = 0 y - 2$

Step 10.2.2.1.3

Multiply $0$ by $y$ .

$f' (0) = 0 - 2$

Step 10.2.2.2

Subtract $2$ from $0$ .

$f' (0) = - 2$

Step 10.2.2.3

The final answer is $- 2$ .

$- 2$

Step 10.3

No local maxima or minima found for $f (x) = 2 x^{2} \cdot (3 x y) + 4 y^{2} - 2 x + 10 y$ .

No local maxima or minima

Step 11