Calculus Examples

$y = x \ln (x + 3)$

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

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 $f (x) = x$ and $g (x) = \ln (x + 3)$ .

$x \frac{d}{d x} [\ln (x + 3)] + \ln (x + 3) \frac{d}{d x} [x]$

Step 1.1.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \ln (x)$ and $g (x) = x + 3$ .

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Step 1.1.2.1

To apply the Chain Rule, set $u$ as $x + 3$ .

$x (\frac{d}{d u} [\ln (u)] \frac{d}{d x} [x + 3]) + \ln (x + 3) \frac{d}{d x} [x]$

Step 1.1.2.2

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

$x (\frac{1}{u} \frac{d}{d x} [x + 3]) + \ln (x + 3) \frac{d}{d x} [x]$

Step 1.1.2.3

Replace all occurrences of $u$ with $x + 3$ .

$x (\frac{1}{x + 3} \frac{d}{d x} [x + 3]) + \ln (x + 3) \frac{d}{d x} [x]$

Step 1.1.3

Differentiate.

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Step 1.1.3.1

Combine $\frac{1}{x + 3}$ and $x$ .

$\frac{x}{x + 3} \frac{d}{d x} [x + 3] + \ln (x + 3) \frac{d}{d x} [x]$

Step 1.1.3.2

By the Sum Rule, the derivative of $x + 3$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [3]$ .

$\frac{x}{x + 3} (\frac{d}{d x} [x] + \frac{d}{d x} [3]) + \ln (x + 3) \frac{d}{d x} [x]$

Step 1.1.3.3

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

$\frac{x}{x + 3} (1 + \frac{d}{d x} [3]) + \ln (x + 3) \frac{d}{d x} [x]$

Step 1.1.3.4

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$\frac{x}{x + 3} (1 + 0) + \ln (x + 3) \frac{d}{d x} [x]$

Step 1.1.3.5

Simplify the expression.

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Step 1.1.3.5.1

Add $1$ and $0$ .

$\frac{x}{x + 3} \cdot 1 + \ln (x + 3) \frac{d}{d x} [x]$

Step 1.1.3.5.2

Multiply $\frac{x}{x + 3}$ by $1$ .

$\frac{x}{x + 3} + \ln (x + 3) \frac{d}{d x} [x]$

Step 1.1.3.6

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

$\frac{x}{x + 3} + \ln (x + 3) \cdot 1$

Step 1.1.3.7

Multiply $\ln (x + 3)$ by $1$ .

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

Step 1.1.4

To write $\ln (x + 3)$ as a fraction with a common denominator, multiply by $\frac{x + 3}{x + 3}$ .

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

Step 1.1.5

Combine the numerators over the common denominator.

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

Step 1.1.6

Simplify.

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Step 1.1.6.1

Simplify the numerator.

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Step 1.1.6.1.1

Simplify each term.

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Step 1.1.6.1.1.1

Apply the distributive property.

$\frac{x + \ln (x + 3) x + \ln (x + 3) \cdot 3}{x + 3}$

Step 1.1.6.1.1.2

Multiply $\ln (x + 3) \cdot 3$ .

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Step 1.1.6.1.1.2.1

Reorder $\ln (x + 3)$ and $3$ .

$\frac{x + \ln (x + 3) x + 3 \cdot \ln (x + 3)}{x + 3}$

Step 1.1.6.1.1.2.2

Simplify $3 \cdot \ln (x + 3)$ by moving $3$ inside the logarithm.

$\frac{x + \ln (x + 3) x + \ln ({(x + 3)}^{3})}{x + 3}$

Step 1.1.6.1.2

Reorder factors in $x + \ln (x + 3) x + \ln ({(x + 3)}^{3})$ .

$\frac{x + x \ln (x + 3) + \ln ({(x + 3)}^{3})}{x + 3}$

Step 1.1.6.2

Reorder terms.

$f' (x) = \frac{x \ln (x + 3) + x + \ln ({(x + 3)}^{3})}{x + 3}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{x \ln (x + 3) + x + \ln ({(x + 3)}^{3})}{x + 3}$ .

$\frac{x \ln (x + 3) + x + \ln ({(x + 3)}^{3})}{x + 3}$

Step 2

Set the first derivative equal to $0$ then solve the equation $\frac{x \ln (x + 3) + x + \ln ({(x + 3)}^{3})}{x + 3} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$\frac{x \ln (x + 3) + x + \ln ({(x + 3)}^{3})}{x + 3} = 0$

Step 2.2

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x \approx - 1.14544928$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

Set the denominator in $\frac{x \ln (x + 3) + x + \ln ({(x + 3)}^{3})}{x + 3}$ equal to $0$ to find where the expression is undefined.

$x + 3 = 0$

Step 3.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 3.3

Set the argument in $\ln (x + 3)$ less than or equal to $0$ to find where the expression is undefined.

$x + 3 \leq 0$

Step 3.4

Subtract $3$ from both sides of the inequality.

$x \leq - 3$

Step 3.5

Set the argument in $\ln ({(x + 3)}^{3})$ less than or equal to $0$ to find where the expression is undefined.

${(x + 3)}^{3} \leq 0$

Step 3.6

Solve for $x$ .

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Step 3.6.1

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

$\sqrt[3]{{(x + 3)}^{3}} \leq \sqrt[3]{0}$

Step 3.6.2

Simplify the equation.

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Step 3.6.2.1

Simplify the left side.

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Step 3.6.2.1.1

Pull terms out from under the radical.

$x + 3 \leq \sqrt[3]{0}$

Step 3.6.2.2

Simplify the right side.

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Step 3.6.2.2.1

Simplify $\sqrt[3]{0}$ .

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Step 3.6.2.2.1.1

Rewrite $0$ as $0^{3}$ .

$x + 3 \leq \sqrt[3]{0^{3}}$

Step 3.6.2.2.1.2

Pull terms out from under the radical.

$x + 3 \leq 0$

Step 3.6.3

Subtract $3$ from both sides of the inequality.

$x \leq - 3$

Step 3.7

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 $0$ .

$x \leq - 3$

$(- \infty, - 3]$

$x \leq - 3$

$(- \infty, - 3]$

Step 4

Evaluate $x \ln (x + 3)$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = - 1.14544928$ .

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Step 4.1.1

Substitute $- 1.14544928$ for $x$ .

$(- 1.14544928) \ln ((- 1.14544928) + 3)$

Step 4.1.2

Simplify.

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Step 4.1.2.1

Add $- 1.14544928$ and $3$ .

$- 1.14544928 \ln (1.85455071)$

Step 4.1.2.2

Simplify $- 1.14544928 \ln (1.85455071)$ by moving $1.14544928$ inside the logarithm.

$- \ln ({1.85455071}^{1.14544928})$

Step 4.1.2.3

Raise $1.85455071$ to the power of $1.14544928$ .

$- \ln (2.02886823)$

Step 4.2

Evaluate at $x = - 3$ .

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Step 4.2.1

Substitute $- 3$ for $x$ .

$(- 3) \ln ((- 3) + 3)$

Step 4.2.2

Simplify.

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Step 4.2.2.1

Add $- 3$ and $3$ .

$- 3 \ln (0)$

Step 4.2.2.2

The natural logarithm of zero is undefined.

Undefined

Step 4.3

List all of the points.

$(- 1.14544928, - \ln (2.02886823))$

Step 5