Calculus Examples

$f (x) = 2 x^{2} \ln (x) - 7 x^{2}$

Step 1

Find the first derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

By the Sum Rule, the derivative of $2 x^{2} \ln (x) - 7 x^{2}$ with respect to $x$ is $\frac{d}{d x} [2 x^{2} \ln (x)] + \frac{d}{d x} [- 7 x^{2}]$ .

$\frac{d}{d x} [2 x^{2} \ln (x)] + \frac{d}{d x} [- 7 x^{2}]$

Step 1.1.2

Evaluate $\frac{d}{d x} [2 x^{2} \ln (x)]$ .

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Step 1.1.2.1

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

$2 \frac{d}{d x} [x^{2} \ln (x)] + \frac{d}{d x} [- 7 x^{2}]$

Step 1.1.2.2

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 $f (x) = x^{2}$ and $g (x) = \ln (x)$ .

$2 (x^{2} \frac{d}{d x} [\ln (x)] + \ln (x) \frac{d}{d x} [x^{2}]) + \frac{d}{d x} [- 7 x^{2}]$

Step 1.1.2.3

The derivative of $\ln (x)$ with respect to $x$ is $\frac{1}{x}$ .

$2 (x^{2} \frac{1}{x} + \ln (x) \frac{d}{d x} [x^{2}]) + \frac{d}{d x} [- 7 x^{2}]$

Step 1.1.2.4

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

$2 (x^{2} \frac{1}{x} + \ln (x) (2 x)) + \frac{d}{d x} [- 7 x^{2}]$

Step 1.1.2.5

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

$2 (\frac{x^{2}}{x} + \ln (x) (2 x)) + \frac{d}{d x} [- 7 x^{2}]$

Step 1.1.2.6

Cancel the common factor of $x^{2}$ and $x$ .

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Step 1.1.2.6.1

Factor $x$ out of $x^{2}$ .

$2 (\frac{x \cdot x}{x} + \ln (x) (2 x)) + \frac{d}{d x} [- 7 x^{2}]$

Step 1.1.2.6.2

Cancel the common factors.

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Step 1.1.2.6.2.1

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

$2 (\frac{x \cdot x}{x^{1}} + \ln (x) (2 x)) + \frac{d}{d x} [- 7 x^{2}]$

Step 1.1.2.6.2.2

Factor $x$ out of $x^{1}$ .

$2 (\frac{x \cdot x}{x \cdot 1} + \ln (x) (2 x)) + \frac{d}{d x} [- 7 x^{2}]$

Step 1.1.2.6.2.3

Cancel the common factor.

$2 (\frac{x \cdot x}{x \cdot 1} + \ln (x) (2 x)) + \frac{d}{d x} [- 7 x^{2}]$

Step 1.1.2.6.2.4

Rewrite the expression.

$2 (\frac{x}{1} + \ln (x) (2 x)) + \frac{d}{d x} [- 7 x^{2}]$

Step 1.1.2.6.2.5

Divide $x$ by $1$ .

$2 (x + \ln (x) (2 x)) + \frac{d}{d x} [- 7 x^{2}]$

Step 1.1.3

Evaluate $\frac{d}{d x} [- 7 x^{2}]$ .

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Step 1.1.3.1

Since $- 7$ is constant with respect to $x$ , the derivative of $- 7 x^{2}$ with respect to $x$ is $- 7 \frac{d}{d x} [x^{2}]$ .

$2 (x + \ln (x) (2 x)) - 7 \frac{d}{d x} [x^{2}]$

Step 1.1.3.2

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

$2 (x + \ln (x) (2 x)) - 7 (2 x)$

Step 1.1.3.3

Multiply $2$ by $- 7$ .

$2 (x + \ln (x) (2 x)) - 14 x$

Step 1.1.4

Simplify.

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Step 1.1.4.1

Apply the distributive property.

$2 x + 2 (\ln (x) (2 x)) - 14 x$

Step 1.1.4.2

Combine terms.

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Step 1.1.4.2.1

Multiply $2$ by $2$ .

$2 x + 4 (\ln (x) (x)) - 14 x$

Step 1.1.4.2.2

Subtract $14 x$ from $2 x$ .

$4 \ln (x) x - 12 x$

Step 1.1.4.3

Reorder terms.

$f' (x) = 4 x \ln (x) - 12 x$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $4 x \ln (x) - 12 x$ .

$4 x \ln (x) - 12 x$

Step 2

Set the first derivative equal to $0$ then solve the equation $4 x \ln (x) - 12 x = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$4 x \ln (x) - 12 x = 0$

Step 2.2

Add $12 x$ to both sides of the equation.

$4 x \ln (x) = 12 x$

Step 2.3

Divide each term in $4 x \ln (x) = 12 x$ by $4 x$ and simplify.

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Step 2.3.1

Divide each term in $4 x \ln (x) = 12 x$ by $4 x$ .

$\frac{4 x \ln (x)}{4 x} = \frac{12 x}{4 x}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $4$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{4 x \ln (x)}{4 x} = \frac{12 x}{4 x}$

Step 2.3.2.1.2

Rewrite the expression.

$\frac{x \ln (x)}{x} = \frac{12 x}{4 x}$

Step 2.3.2.2

Cancel the common factor of $x$ .

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Step 2.3.2.2.1

Cancel the common factor.

$\frac{x \ln (x)}{x} = \frac{12 x}{4 x}$

Step 2.3.2.2.2

Divide $\ln (x)$ by $1$ .

$\ln (x) = \frac{12 x}{4 x}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Cancel the common factor of $12$ and $4$ .

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Step 2.3.3.1.1

Factor $4$ out of $12 x$ .

$\ln (x) = \frac{4 (3 x)}{4 x}$

Step 2.3.3.1.2

Cancel the common factors.

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Step 2.3.3.1.2.1

Factor $4$ out of $4 x$ .

$\ln (x) = \frac{4 (3 x)}{4 (x)}$

Step 2.3.3.1.2.2

Cancel the common factor.

$\ln (x) = \frac{4 (3 x)}{4 x}$

Step 2.3.3.1.2.3

Rewrite the expression.

$\ln (x) = \frac{3 x}{x}$

Step 2.3.3.2

Cancel the common factor of $x$ .

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Step 2.3.3.2.1

Cancel the common factor.

$\ln (x) = \frac{3 x}{x}$

Step 2.3.3.2.2

Divide $3$ by $1$ .

$\ln (x) = 3$

Step 2.4

To solve for $x$ , rewrite the equation using properties of logarithms.

$e^{\ln (x)} = e^{3}$

Step 2.5

Rewrite $\ln (x) = 3$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$e^{3} = x$

Step 2.6

Rewrite the equation as $x = e^{3}$ .

$x = e^{3}$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

Set the argument in $\ln (x)$ less than or equal to $0$ to find where the expression is undefined.

$x \leq 0$

Step 3.2

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 $0$ .

$x \leq 0$

$(- \infty, 0]$

$x \leq 0$

$(- \infty, 0]$

Step 4

Evaluate $2 x^{2} \ln (x) - 7 x^{2}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = e^{3}$ .

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Step 4.1.1

Substitute $e^{3}$ for $x$ .

$2 {(e^{3})}^{2} \ln (e^{3}) - 7 {(e^{3})}^{2}$

Step 4.1.2

Simplify.

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Step 4.1.2.1

Simplify each term.

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Step 4.1.2.1.1

Multiply the exponents in ${(e^{3})}^{2}$ .

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Step 4.1.2.1.1.1

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

$2 e^{3 \cdot 2} \ln (e^{3}) - 7 {(e^{3})}^{2}$

Step 4.1.2.1.1.2

Multiply $3$ by $2$ .

$2 e^{6} \ln (e^{3}) - 7 {(e^{3})}^{2}$

Step 4.1.2.1.2

Use logarithm rules to move $3$ out of the exponent.

$2 e^{6} (3 \ln (e)) - 7 {(e^{3})}^{2}$

Step 4.1.2.1.3

The natural logarithm of $e$ is $1$ .

$2 e^{6} (3 \cdot 1) - 7 {(e^{3})}^{2}$

Step 4.1.2.1.4

Multiply $3$ by $1$ .

$2 e^{6} \cdot 3 - 7 {(e^{3})}^{2}$

Step 4.1.2.1.5

Multiply $3$ by $2$ .

$6 e^{6} - 7 {(e^{3})}^{2}$

Step 4.1.2.1.6

Multiply the exponents in ${(e^{3})}^{2}$ .

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Step 4.1.2.1.6.1

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

$6 e^{6} - 7 e^{3 \cdot 2}$

Step 4.1.2.1.6.2

Multiply $3$ by $2$ .

$6 e^{6} - 7 e^{6}$

Step 4.1.2.2

Subtract $7 e^{6}$ from $6 e^{6}$ .

$- e^{6}$

Step 4.2

Evaluate at $x = 0$ .

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Step 4.2.1

Substitute $0$ for $x$ .

$2 {(0)}^{2} \ln (0) - 7 {(0)}^{2}$

Step 4.2.2

The natural logarithm of zero is undefined.

Undefined

Step 4.3

List all of the points.

$(e^{3}, - e^{6})$

Step 5