Calculus Examples

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

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 Constant Multiple Rule.

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Step 1.1.1.1

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

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

Step 1.1.1.2

Since $- 1$ is constant with respect to $x$ , the derivative of $- {(x + 3)}^{\frac{1}{2}}$ with respect to $x$ is $- \frac{d}{d x} [{(x + 3)}^{\frac{1}{2}}]$ .

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

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

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Step 1.1.2.1

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

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

Step 1.1.2.2

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

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

Step 1.1.2.3

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

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

Step 1.1.3

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

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

Step 1.1.4

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

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

Step 1.1.5

Combine the numerators over the common denominator.

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

Step 1.1.6

Simplify the numerator.

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Step 1.1.6.1

Multiply $- 1$ by $2$ .

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

Step 1.1.6.2

Subtract $2$ from $1$ .

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

Step 1.1.7

Combine fractions.

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Step 1.1.7.1

Move the negative in front of the fraction.

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

Step 1.1.7.2

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

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

Step 1.1.7.3

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

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

Step 1.1.8

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

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

Step 1.1.9

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

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

Step 1.1.10

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

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

Step 1.1.11

Simplify the expression.

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Step 1.1.11.1

Add $1$ and $0$ .

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

Step 1.1.11.2

Multiply $- 1$ by $1$ .

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

Step 1.2

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

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

Step 2

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

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Step 2.1

Set the first derivative equal to $0$ .

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

Step 2.2

Set the numerator equal to zero.

$1 = 0$

Step 2.3

Since $1 \neq 0$ , there are no solutions.

No solution

Step 3

There are no values of $x$ in the domain of the original problem where the derivative is $0$ or undefined.

No critical points found

Step 4

Find where the derivative is undefined.

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Step 4.1

Convert expressions with fractional exponents to radicals.

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Step 4.1.1

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

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

Step 4.1.2

Anything raised to $1$ is the base itself.

$- \frac{1}{2 \sqrt{x + 3}}$

Step 4.2

Set the denominator in $\frac{1}{2 \sqrt{x + 3}}$ equal to $0$ to find where the expression is undefined.

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

Step 4.3

Solve for $x$ .

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Step 4.3.1

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 4.3.2

Simplify each side of the equation.

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Step 4.3.2.1

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 4.3.2.2

Simplify the left side.

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Step 4.3.2.2.1

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

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Step 4.3.2.2.1.1

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

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

Step 4.3.2.2.1.2

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

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

Step 4.3.2.2.1.3

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

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Step 4.3.2.2.1.3.1

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

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

Step 4.3.2.2.1.3.2

Cancel the common factor of $2$ .

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Step 4.3.2.2.1.3.2.1

Cancel the common factor.

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

Step 4.3.2.2.1.3.2.2

Rewrite the expression.

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

Step 4.3.2.2.1.4

Simplify.

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

Step 4.3.2.2.1.5

Apply the distributive property.

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

Step 4.3.2.2.1.6

Multiply $4$ by $3$ .

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

Step 4.3.2.3

Simplify the right side.

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Step 4.3.2.3.1

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

$4 x + 12 = 0$

Step 4.3.3

Solve for $x$ .

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Step 4.3.3.1

Subtract $12$ from both sides of the equation.

$4 x = - 12$

Step 4.3.3.2

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

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Step 4.3.3.2.1

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

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

Step 4.3.3.2.2

Simplify the left side.

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Step 4.3.3.2.2.1

Cancel the common factor of $4$ .

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Step 4.3.3.2.2.1.1

Cancel the common factor.

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

Step 4.3.3.2.2.1.2

Divide $x$ by $1$ .

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

Step 4.3.3.2.3

Simplify the right side.

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Step 4.3.3.2.3.1

Divide $- 12$ by $4$ .

$x = - 3$

Step 4.4

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

$x + 3 < 0$

Step 4.5

Subtract $3$ from both sides of the inequality.

$x < - 3$

Step 4.6

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 5

After finding the point that makes the derivative $f' (x) = - \frac{1}{2 {(x + 3)}^{\frac{1}{2}}}$ equal to $0$ or undefined, the interval to check where $f (x) = - \sqrt{x + 3}$ is increasing and where it is decreasing is $(- \infty, - 3) \cup (- 3, \infty)$ .

$(- \infty, - 3) \cup (- 3, \infty)$

Step 6

Substitute a value from the interval $(- \infty, - 3)$ into the derivative to determine if the function is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $- 4$ in the expression.

$f' (- 4) = - \frac{1}{2 {((- 4) + 3)}^{\frac{1}{2}}}$

Step 6.2

Simplify the result.

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Step 6.2.1

Simplify the denominator.

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Step 6.2.1.1

Add $- 4$ and $3$ .

$f' (- 4) = - \frac{1}{2 {(- 1)}^{\frac{1}{2}}}$

Step 6.2.1.2

Rewrite ${(- 1)}^{\frac{1}{2}}$ as $\sqrt{{(- 1)}^{1}}$ .

$f' (- 4) = - \frac{1}{2 \sqrt{- 1}}$

Step 6.2.1.3

Evaluate the exponent.

$f' (- 4) = - \frac{1}{2 \sqrt{- 1}}$

Step 6.2.1.4

Rewrite $\sqrt{- 1}$ as $i$ .

$f' (- 4) = - \frac{1}{2 i}$

Step 6.2.2

Multiply the numerator and denominator of $\frac{1}{2 i}$ by the conjugate of $2 i$ to make the denominator real.

$f' (- 4) = - (\frac{1}{2 i} \cdot \frac{i}{i})$

Step 6.2.3

Multiply.

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Step 6.2.3.1

Combine.

$f' (- 4) = - \frac{1 i}{2 i i}$

Step 6.2.3.2

Multiply $i$ by $1$ .

$f' (- 4) = - \frac{i}{2 i i}$

Step 6.2.3.3

Simplify the denominator.

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Step 6.2.3.3.1

Add parentheses.

$f' (- 4) = - \frac{i}{2 (i i)}$

Step 6.2.3.3.2

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

$f' (- 4) = - \frac{i}{2 (i i)}$

Step 6.2.3.3.3

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

$f' (- 4) = - \frac{i}{2 (i i)}$

Step 6.2.3.3.4

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

$f' (- 4) = - \frac{i}{2 i^{1 + 1}}$

Step 6.2.3.3.5

Add $1$ and $1$ .

$f' (- 4) = - \frac{i}{2 i^{2}}$

Step 6.2.3.3.6

Rewrite $i^{2}$ as $- 1$ .

$f' (- 4) = - \frac{i}{2 \cdot - 1}$

Step 6.2.4

Multiply $2$ by $- 1$ .

$f' (- 4) = - \frac{i}{- 2}$

Step 6.2.5

Move the negative in front of the fraction.

$f' (- 4) = \frac{i}{2}$

Step 6.2.6

The final answer is $\frac{i}{2}$ .

$\frac{i}{2}$

Step 6.3

At $x = - 4$ the derivative is $\frac{i}{2}$ . Since this contains an imaginary number, the function does not exist on $(- \infty, - 3)$ .

Function is not real on $(- \infty, - 3)$ since $f' (x)$ is imaginary

Step 7

Substitute a value from the interval $(- 3, \infty)$ into the derivative to determine if the function is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $- 2$ in the expression.

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

Step 7.2

Simplify the result.

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Step 7.2.1

Simplify the denominator.

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Step 7.2.1.1

Add $- 2$ and $3$ .

$f' (- 2) = - \frac{1}{2 \cdot 1^{\frac{1}{2}}}$

Step 7.2.1.2

One to any power is one.

$f' (- 2) = - \frac{1}{2 \cdot 1}$

Step 7.2.2

Multiply $2$ by $1$ .

$f' (- 2) = - \frac{1}{2}$

Step 7.2.3

The final answer is $- \frac{1}{2}$ .

$- \frac{1}{2}$

Step 7.3

At $x = - 2$ the derivative is $- \frac{1}{2}$ . Since this is negative, the function is decreasing on $(- 3, \infty)$ .

Decreasing on $(- 3, \infty)$ since $f' (x) < 0$

Step 8

List the intervals on which the function is increasing and decreasing.

Decreasing on: $(- 3, \infty)$

Step 9