Calculus Examples

$f (x) = \sqrt{x^{3} + 3 x^{2}}$

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

Rewrite $\frac{d}{d x} [\sqrt{x^{3} + 3 x^{2}}]$ as $\frac{d}{d x} [x \sqrt{x + 3}]$ .

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Step 1.1.1.1

Rewrite $x^{3} + 3 x^{2}$ as $x^{2} (x + 3)$ .

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Step 1.1.1.1.1

Factor $x^{2}$ out of $x^{3}$ .

$\frac{d}{d x} [\sqrt{x^{2} x + 3 x^{2}}]$

Step 1.1.1.1.2

Factor $x^{2}$ out of $3 x^{2}$ .

$\frac{d}{d x} [\sqrt{x^{2} x + x^{2} \cdot 3}]$

Step 1.1.1.1.3

Factor $x^{2}$ out of $x^{2} x + x^{2} \cdot 3$ .

$\frac{d}{d x} [\sqrt{x^{2} (x + 3)}]$

Step 1.1.1.2

Pull terms out from under the radical.

$\frac{d}{d x} [x \sqrt{x + 3}]$

Step 1.1.2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 3}$ as ${(x + 3)}^{\frac{1}{2}}$ .

$\frac{d}{d x} [x {(x + 3)}^{\frac{1}{2}}]$

Step 1.1.3

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 $g (x) = {(x + 3)}^{\frac{1}{2}}$ .

$x \frac{d}{d x} [{(x + 3)}^{\frac{1}{2}}] + {(x + 3)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.1.4

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = x + 3$ .

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Step 1.1.4.1

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

$x (\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [x + 3]) + {(x + 3)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.1.4.2

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

$x (\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [x + 3]) + {(x + 3)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.1.4.3

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

$x (\frac{1}{2} {(x + 3)}^{\frac{1}{2} - 1} \frac{d}{d x} [x + 3]) + {(x + 3)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.1.5

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x (\frac{1}{2} {(x + 3)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [x + 3]) + {(x + 3)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.1.6

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

$x (\frac{1}{2} {(x + 3)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [x + 3]) + {(x + 3)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.1.7

Combine the numerators over the common denominator.

$x (\frac{1}{2} {(x + 3)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [x + 3]) + {(x + 3)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.1.8

Simplify the numerator.

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Step 1.1.8.1

Multiply $- 1$ by $2$ .

$x (\frac{1}{2} {(x + 3)}^{\frac{1 - 2}{2}} \frac{d}{d x} [x + 3]) + {(x + 3)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.1.8.2

Subtract $2$ from $1$ .

$x (\frac{1}{2} {(x + 3)}^{\frac{- 1}{2}} \frac{d}{d x} [x + 3]) + {(x + 3)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.1.9

Combine fractions.

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Step 1.1.9.1

Move the negative in front of the fraction.

$x (\frac{1}{2} {(x + 3)}^{- \frac{1}{2}} \frac{d}{d x} [x + 3]) + {(x + 3)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.1.9.2

Combine $\frac{1}{2}$ and ${(x + 3)}^{- \frac{1}{2}}$ .

$x (\frac{{(x + 3)}^{- \frac{1}{2}}}{2} \frac{d}{d x} [x + 3]) + {(x + 3)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.1.9.3

Move ${(x + 3)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$x (\frac{1}{2 {(x + 3)}^{\frac{1}{2}}} \frac{d}{d x} [x + 3]) + {(x + 3)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.1.9.4

Combine $\frac{1}{2 {(x + 3)}^{\frac{1}{2}}}$ and $x$ .

$\frac{x}{2 {(x + 3)}^{\frac{1}{2}}} \frac{d}{d x} [x + 3] + {(x + 3)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.1.10

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

$\frac{x}{2 {(x + 3)}^{\frac{1}{2}}} (\frac{d}{d x} [x] + \frac{d}{d x} [3]) + {(x + 3)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.1.11

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

$\frac{x}{2 {(x + 3)}^{\frac{1}{2}}} (1 + \frac{d}{d x} [3]) + {(x + 3)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.1.12

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

$\frac{x}{2 {(x + 3)}^{\frac{1}{2}}} (1 + 0) + {(x + 3)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.1.13

Simplify the expression.

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Step 1.1.13.1

Add $1$ and $0$ .

$\frac{x}{2 {(x + 3)}^{\frac{1}{2}}} \cdot 1 + {(x + 3)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.1.13.2

Multiply $\frac{x}{2 {(x + 3)}^{\frac{1}{2}}}$ by $1$ .

$\frac{x}{2 {(x + 3)}^{\frac{1}{2}}} + {(x + 3)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.1.14

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

$\frac{x}{2 {(x + 3)}^{\frac{1}{2}}} + {(x + 3)}^{\frac{1}{2}} \cdot 1$

Step 1.1.15

Multiply ${(x + 3)}^{\frac{1}{2}}$ by $1$ .

$\frac{x}{2 {(x + 3)}^{\frac{1}{2}}} + {(x + 3)}^{\frac{1}{2}}$

Step 1.1.16

To write ${(x + 3)}^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{2 {(x + 3)}^{\frac{1}{2}}}{2 {(x + 3)}^{\frac{1}{2}}}$ .

$\frac{x}{2 {(x + 3)}^{\frac{1}{2}}} + {(x + 3)}^{\frac{1}{2}} \cdot \frac{2 {(x + 3)}^{\frac{1}{2}}}{2 {(x + 3)}^{\frac{1}{2}}}$

Step 1.1.17

Combine ${(x + 3)}^{\frac{1}{2}}$ and $\frac{2 {(x + 3)}^{\frac{1}{2}}}{2 {(x + 3)}^{\frac{1}{2}}}$ .

$\frac{x}{2 {(x + 3)}^{\frac{1}{2}}} + \frac{{(x + 3)}^{\frac{1}{2}} (2 {(x + 3)}^{\frac{1}{2}})}{2 {(x + 3)}^{\frac{1}{2}}}$

Step 1.1.18

Combine the numerators over the common denominator.

$\frac{x + {(x + 3)}^{\frac{1}{2}} (2 {(x + 3)}^{\frac{1}{2}})}{2 {(x + 3)}^{\frac{1}{2}}}$

Step 1.1.19

Multiply ${(x + 3)}^{\frac{1}{2}}$ by ${(x + 3)}^{\frac{1}{2}}$ by adding the exponents.

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Step 1.1.19.1

Move ${(x + 3)}^{\frac{1}{2}}$ .

$\frac{x + {(x + 3)}^{\frac{1}{2}} {(x + 3)}^{\frac{1}{2}} \cdot 2}{2 {(x + 3)}^{\frac{1}{2}}}$

Step 1.1.19.2

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

$\frac{x + {(x + 3)}^{\frac{1}{2} + \frac{1}{2}} \cdot 2}{2 {(x + 3)}^{\frac{1}{2}}}$

Step 1.1.19.3

Combine the numerators over the common denominator.

$\frac{x + {(x + 3)}^{\frac{1 + 1}{2}} \cdot 2}{2 {(x + 3)}^{\frac{1}{2}}}$

Step 1.1.19.4

Add $1$ and $1$ .

$\frac{x + {(x + 3)}^{\frac{2}{2}} \cdot 2}{2 {(x + 3)}^{\frac{1}{2}}}$

Step 1.1.19.5

Divide $2$ by $2$ .

$\frac{x + {(x + 3)}^{1} \cdot 2}{2 {(x + 3)}^{\frac{1}{2}}}$

Step 1.1.20

Simplify ${(x + 3)}^{1} \cdot 2$ .

$\frac{x + (x + 3) \cdot 2}{2 {(x + 3)}^{\frac{1}{2}}}$

Step 1.1.21

Move $2$ to the left of $x + 3$ .

$\frac{x + 2 \cdot (x + 3)}{2 {(x + 3)}^{\frac{1}{2}}}$

Step 1.1.22

Simplify.

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Step 1.1.22.1

Apply the distributive property.

$\frac{x + 2 x + 2 \cdot 3}{2 {(x + 3)}^{\frac{1}{2}}}$

Step 1.1.22.2

Simplify the numerator.

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Step 1.1.22.2.1

Multiply $2$ by $3$ .

$\frac{x + 2 x + 6}{2 {(x + 3)}^{\frac{1}{2}}}$

Step 1.1.22.2.2

Add $x$ and $2 x$ .

$\frac{3 x + 6}{2 {(x + 3)}^{\frac{1}{2}}}$

Step 1.1.22.3

Factor $3$ out of $3 x + 6$ .

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Step 1.1.22.3.1

Factor $3$ out of $3 x$ .

$\frac{3 (x) + 6}{2 {(x + 3)}^{\frac{1}{2}}}$

Step 1.1.22.3.2

Factor $3$ out of $6$ .

$\frac{3 x + 3 \cdot 2}{2 {(x + 3)}^{\frac{1}{2}}}$

Step 1.1.22.3.3

Factor $3$ out of $3 x + 3 \cdot 2$ .

$f' (x) = \frac{3 (x + 2)}{2 {(x + 3)}^{\frac{1}{2}}}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{3 (x + 2)}{2 {(x + 3)}^{\frac{1}{2}}}$ .

$\frac{3 (x + 2)}{2 {(x + 3)}^{\frac{1}{2}}}$

Step 2

Set the first derivative equal to $0$ then solve the equation $\frac{3 (x + 2)}{2 {(x + 3)}^{\frac{1}{2}}} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$\frac{3 (x + 2)}{2 {(x + 3)}^{\frac{1}{2}}} = 0$

Step 2.2

Set the numerator equal to zero.

$3 (x + 2) = 0$

Step 2.3

Solve the equation for $x$ .

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Step 2.3.1

Divide each term in $3 (x + 2) = 0$ by $3$ and simplify.

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Step 2.3.1.1

Divide each term in $3 (x + 2) = 0$ by $3$ .

$\frac{3 (x + 2)}{3} = \frac{0}{3}$

Step 2.3.1.2

Simplify the left side.

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Step 2.3.1.2.1

Cancel the common factor of $3$ .

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Step 2.3.1.2.1.1

Cancel the common factor.

$\frac{3 (x + 2)}{3} = \frac{0}{3}$

Step 2.3.1.2.1.2

Divide $x + 2$ by $1$ .

$x + 2 = \frac{0}{3}$

Step 2.3.1.3

Simplify the right side.

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Step 2.3.1.3.1

Divide $0$ by $3$ .

$x + 2 = 0$

Step 2.3.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

Convert expressions with fractional exponents to radicals.

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Step 3.1.1

Apply the rule $x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ to rewrite the exponentiation as a radical.

$\frac{3 (x + 2)}{2 \sqrt{{(x + 3)}^{1}}}$

Step 3.1.2

Anything raised to $1$ is the base itself.

$\frac{3 (x + 2)}{2 \sqrt{x + 3}}$

Step 3.2

Set the denominator in $\frac{3 (x + 2)}{2 \sqrt{x + 3}}$ equal to $0$ to find where the expression is undefined.

$2 \sqrt{x + 3} = 0$

Step 3.3

Solve for $x$ .

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Step 3.3.1

To remove the radical on the left side of the equation, square both sides of the equation.

${(2 \sqrt{x + 3})}^{2} = 0^{2}$

Step 3.3.2

Simplify each side of the equation.

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Step 3.3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 3}$ as ${(x + 3)}^{\frac{1}{2}}$ .

${(2 {(x + 3)}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 3.3.2.2

Simplify the left side.

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Step 3.3.2.2.1

Simplify ${(2 {(x + 3)}^{\frac{1}{2}})}^{2}$ .

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Step 3.3.2.2.1.1

Apply the product rule to $2 {(x + 3)}^{\frac{1}{2}}$ .

$2^{2} {({(x + 3)}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 3.3.2.2.1.2

Raise $2$ to the power of $2$ .

$4 {({(x + 3)}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 3.3.2.2.1.3

Multiply the exponents in ${({(x + 3)}^{\frac{1}{2}})}^{2}$ .

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Step 3.3.2.2.1.3.1

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

$4 {(x + 3)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 3.3.2.2.1.3.2

Cancel the common factor of $2$ .

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Step 3.3.2.2.1.3.2.1

Cancel the common factor.

$4 {(x + 3)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 3.3.2.2.1.3.2.2

Rewrite the expression.

$4 {(x + 3)}^{1} = 0^{2}$

Step 3.3.2.2.1.4

Simplify.

$4 (x + 3) = 0^{2}$

Step 3.3.2.2.1.5

Apply the distributive property.

$4 x + 4 \cdot 3 = 0^{2}$

Step 3.3.2.2.1.6

Multiply $4$ by $3$ .

$4 x + 12 = 0^{2}$

Step 3.3.2.3

Simplify the right side.

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Step 3.3.2.3.1

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

$4 x + 12 = 0$

Step 3.3.3

Solve for $x$ .

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Step 3.3.3.1

Subtract $12$ from both sides of the equation.

$4 x = - 12$

Step 3.3.3.2

Divide each term in $4 x = - 12$ by $4$ and simplify.

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Step 3.3.3.2.1

Divide each term in $4 x = - 12$ by $4$ .

$\frac{4 x}{4} = \frac{- 12}{4}$

Step 3.3.3.2.2

Simplify the left side.

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Step 3.3.3.2.2.1

Cancel the common factor of $4$ .

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Step 3.3.3.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{- 12}{4}$

Step 3.3.3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 12}{4}$

Step 3.3.3.2.3

Simplify the right side.

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Step 3.3.3.2.3.1

Divide $- 12$ by $4$ .

$x = - 3$

Step 3.4

Set the radicand in $\sqrt{x + 3}$ less than $0$ to find where the expression is undefined.

$x + 3 < 0$

Step 3.5

Subtract $3$ from both sides of the inequality.

$x < - 3$

Step 3.6

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 $\sqrt{x^{3} + 3 x^{2}}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = - 2$ .

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Step 4.1.1

Substitute $- 2$ for $x$ .

$\sqrt{{(- 2)}^{3} + 3 {(- 2)}^{2}}$

Step 4.1.2

Simplify.

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Step 4.1.2.1

Raise $- 2$ to the power of $3$ .

$\sqrt{- 8 + 3 {(- 2)}^{2}}$

Step 4.1.2.2

Raise $- 2$ to the power of $2$ .

$\sqrt{- 8 + 3 \cdot 4}$

Step 4.1.2.3

Multiply $3$ by $4$ .

$\sqrt{- 8 + 12}$

Step 4.1.2.4

Add $- 8$ and $12$ .

$\sqrt{4}$

Step 4.1.2.5

Rewrite $4$ as $2^{2}$ .

$\sqrt{2^{2}}$

Step 4.1.2.6

Pull terms out from under the radical, assuming positive real numbers.

$2$

Step 4.2

Evaluate at $x = - 3$ .

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Step 4.2.1

Substitute $- 3$ for $x$ .

$\sqrt{{(- 3)}^{3} + 3 {(- 3)}^{2}}$

Step 4.2.2

Simplify.

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Step 4.2.2.1

Raise $- 3$ to the power of $3$ .

$\sqrt{- 27 + 3 {(- 3)}^{2}}$

Step 4.2.2.2

Raise $- 3$ to the power of $2$ .

$\sqrt{- 27 + 3 \cdot 9}$

Step 4.2.2.3

Multiply $3$ by $9$ .

$\sqrt{- 27 + 27}$

Step 4.2.2.4

Add $- 27$ and $27$ .

$\sqrt{0}$

Step 4.2.2.5

Rewrite $0$ as $0^{2}$ .

$\sqrt{0^{2}}$

Step 4.2.2.6

Pull terms out from under the radical, assuming positive real numbers.

$0$

Step 4.3

List all of the points.

$(- 2, 2), (- 3, 0)$

Step 5