Find Where Increasing/Decreasing Using Derivatives f(x) = square root of 100-x^2

$f (x) = \sqrt{100 - x^{2}}$

Step 1

Find the first derivative.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{100 - x^{2}}$ as ${(100 - x^{2})}^{\frac{1}{2}}$ .

$f' (x) = \frac{d}{d x} ({(100 - x^{2})}^{\frac{1}{2}})$

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 $g (x) = 100 - x^{2}$ .

To apply the Chain Rule, set $u$ as $100 - x^{2}$ .

$f' (x) = \frac{d}{d u} (u^{\frac{1}{2}}) \frac{d}{d x} (100 - x^{2})$

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

$f' (x) = \frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} (100 - x^{2})$

Replace all occurrences of $u$ with $100 - x^{2}$ .

$f' (x) = \frac{1}{2} \cdot {(100 - x^{2})}^{\frac{1}{2} - 1} \frac{d}{d x} (100 - x^{2})$

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

$f' (x) = \frac{1}{2} \cdot {(100 - x^{2})}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} (100 - x^{2})$

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

$f' (x) = \frac{1}{2} \cdot {(100 - x^{2})}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} (100 - x^{2})$

Combine the numerators over the common denominator.

$f' (x) = \frac{1}{2} \cdot {(100 - x^{2})}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} (100 - x^{2})$

Simplify the numerator.

Multiply $- 1$ by $2$ .

$f' (x) = \frac{1}{2} \cdot {(100 - x^{2})}^{\frac{1 - 2}{2}} \frac{d}{d x} (100 - x^{2})$

Subtract $2$ from $1$ .

$f' (x) = \frac{1}{2} \cdot {(100 - x^{2})}^{\frac{- 1}{2}} \frac{d}{d x} (100 - x^{2})$

Combine fractions.

Move the negative in front of the fraction.

$f' (x) = \frac{1}{2} \cdot {(100 - x^{2})}^{- \frac{1}{2}} \frac{d}{d x} (100 - x^{2})$

Combine $\frac{1}{2}$ and ${(100 - x^{2})}^{- \frac{1}{2}}$ .

$f' (x) = \frac{{(100 - x^{2})}^{- \frac{1}{2}}}{2} \cdot \frac{d}{d x} (100 - x^{2})$

Move ${(100 - x^{2})}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f' (x) = \frac{1}{2 {(100 - x^{2})}^{\frac{1}{2}}} \cdot \frac{d}{d x} (100 - x^{2})$

By the Sum Rule, the derivative of $100 - x^{2}$ with respect to $x$ is $\frac{d}{d x} [100] + \frac{d}{d x} [- x^{2}]$ .

$f' (x) = \frac{1}{2 {(100 - x^{2})}^{\frac{1}{2}}} \cdot (\frac{d}{d x} (100) + \frac{d}{d x} (- x^{2}))$

Since $100$ is constant with respect to $x$ , the derivative of $100$ with respect to $x$ is $0$ .

$f' (x) = \frac{1}{2 {(100 - x^{2})}^{\frac{1}{2}}} \cdot (0 + \frac{d}{d x} (- x^{2}))$

Add $0$ and $\frac{d}{d x} [- x^{2}]$ .

$f' (x) = \frac{1}{2 {(100 - x^{2})}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- x^{2})$

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

$f' (x) = \frac{1}{2 {(100 - x^{2})}^{\frac{1}{2}}} \cdot (- \frac{d}{d x} x^{2})$

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

$f' (x) = \frac{1}{2 {(100 - x^{2})}^{\frac{1}{2}}} \cdot (- (2 x))$

Simplify terms.

Multiply $2$ by $- 1$ .

$f' (x) = \frac{1}{2 {(100 - x^{2})}^{\frac{1}{2}}} \cdot (- 2 x)$

Combine $- 2$ and $\frac{1}{2 {(100 - x^{2})}^{\frac{1}{2}}}$ .

$f' (x) = \frac{- 2}{2 {(100 - x^{2})}^{\frac{1}{2}}} \cdot x$

Combine $\frac{- 2}{2 {(100 - x^{2})}^{\frac{1}{2}}}$ and $x$ .

$f' (x) = \frac{- 2 x}{2 {(100 - x^{2})}^{\frac{1}{2}}}$

Factor $2$ out of $- 2 x$ .

$f' (x) = \frac{2 (- x)}{2 {(100 - x^{2})}^{\frac{1}{2}}}$

Cancel the common factors.

Factor $2$ out of $2 {(100 - x^{2})}^{\frac{1}{2}}$ .

$f' (x) = \frac{2 (- x)}{2 ({(100 - x^{2})}^{\frac{1}{2}})}$

Cancel the common factor.

$f' (x) = \frac{2 (- x)}{2 {(100 - x^{2})}^{\frac{1}{2}}}$

Rewrite the expression.

$f' (x) = \frac{- x}{{(100 - x^{2})}^{\frac{1}{2}}}$

Move the negative in front of the fraction.

$f' (x) = - \frac{x}{{(100 - x^{2})}^{\frac{1}{2}}}$

The first derivative of $f (x)$ with respect to $x$ is $- \frac{x}{{(100 - x^{2})}^{\frac{1}{2}}}$ .

$- \frac{x}{{(100 - x^{2})}^{\frac{1}{2}}}$

Step 2

Set the first derivative equal to $0$ then solve the equation $- \frac{x}{{(100 - x^{2})}^{\frac{1}{2}}} = 0$ .

Set the first derivative equal to $0$ .

$- \frac{x}{{(100 - x^{2})}^{\frac{1}{2}}} = 0$

Set the numerator equal to zero.

$x = 0$

Step 3

The values which make the derivative equal to $0$ are $0$ .

$0$

Step 4

Find where the derivative is undefined.

Convert expressions with fractional exponents to radicals.

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

$- \frac{x}{\sqrt{{(100 - x^{2})}^{1}}}$

Anything raised to $1$ is the base itself.

$- \frac{x}{\sqrt{100 - x^{2}}}$

Set the denominator in $\frac{x}{\sqrt{100 - x^{2}}}$ equal to $0$ to find where the expression is undefined.

$\sqrt{100 - x^{2}} = 0$

Solve for $x$ .

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

${\sqrt{100 - x^{2}}}^{2} = 0^{2}$

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{100 - x^{2}}$ as ${(100 - x^{2})}^{\frac{1}{2}}$ .

${({(100 - x^{2})}^{\frac{1}{2}})}^{2} = 0^{2}$

Simplify the left side.

Simplify ${({(100 - x^{2})}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(100 - x^{2})}^{\frac{1}{2}})}^{2}$ .

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

${(100 - x^{2})}^{\frac{1}{2} \cdot 2} = 0^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(100 - x^{2})}^{\frac{1}{2} \cdot 2} = 0^{2}$

Rewrite the expression.

${(100 - x^{2})}^{1} = 0^{2}$

Simplify.

$100 - x^{2} = 0^{2}$

Simplify the right side.

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

$100 - x^{2} = 0$

Solve for $x$ .

Subtract $100$ from both sides of the equation.

$- x^{2} = - 100$

Divide each term in $- x^{2} = - 100$ by $- 1$ and simplify.

Divide each term in $- x^{2} = - 100$ by $- 1$ .

$\frac{- x^{2}}{- 1} = \frac{- 100}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x^{2}}{1} = \frac{- 100}{- 1}$

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

$x^{2} = \frac{- 100}{- 1}$

Simplify the right side.

Divide $- 100$ by $- 1$ .

$x^{2} = 100$

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

$x = \pm \sqrt{100}$

Simplify $\pm \sqrt{100}$ .

Rewrite $100$ as $10^{2}$ .

$x = \pm \sqrt{10^{2}}$

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

$x = \pm 10$

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

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

$x = 10$

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

$x = - 10$

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

$x = 10, - 10$

Set the radicand in $\sqrt{100 - x^{2}}$ less than $0$ to find where the expression is undefined.

$100 - x^{2} < 0$

Solve for $x$ .

Subtract $100$ from both sides of the inequality.

$- x^{2} < - 100$

Divide each term in $- x^{2} < - 100$ by $- 1$ and simplify.

Divide each term in $- x^{2} < - 100$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x^{2}}{- 1} > \frac{- 100}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x^{2}}{1} > \frac{- 100}{- 1}$

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

$x^{2} > \frac{- 100}{- 1}$

Simplify the right side.

Divide $- 100$ by $- 1$ .

$x^{2} > 100$

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

$\sqrt{x^{2}} > \sqrt{100}$

Simplify the equation.

Simplify the left side.

Pull terms out from under the radical.

$| x | > \sqrt{100}$

Simplify the right side.

Simplify $\sqrt{100}$ .

Rewrite $100$ as $10^{2}$ .

$| x | > \sqrt{10^{2}}$

Pull terms out from under the radical.

$| x | > | 10 |$

The absolute value is the distance between a number and zero. The distance between $0$ and $10$ is $10$ .

$| x | > 10$

Write $| x | > 10$ as a piecewise.

To find the interval for the first piece, find where the inside of the absolute value is non-negative.

$x \geq 0$

In the piece where $x$ is non-negative, remove the absolute value.

$x > 10$

To find the interval for the second piece, find where the inside of the absolute value is negative.

$x < 0$

In the piece where $x$ is negative, remove the absolute value and multiply by $- 1$ .

$- x > 10$

Write as a piecewise.

${\begin{cases} x > 10 & x \geq 0 \\ - x > 10 & x < 0 \end{cases}$

Find the intersection of $x > 10$ and $x \geq 0$ .

$x > 10$

Divide each term in $- x > 10$ by $- 1$ and simplify.

Divide each term in $- x > 10$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} < \frac{10}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} < \frac{10}{- 1}$

Divide $x$ by $1$ .

$x < \frac{10}{- 1}$

Simplify the right side.

Divide $10$ by $- 1$ .

$x < - 10$

Find the union of the solutions.

$x < - 10$ or $x > 10$

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 $0$ .

$x \leq - 10, x \geq 10$

$(- \infty, - 10] \cup [10, \infty)$

$x \leq - 10, x \geq 10$

$(- \infty, - 10] \cup [10, \infty)$

Step 5

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

$(- \infty, - 10) \cup (- 10, 0) \cup (0, 10) \cup (10, \infty)$

Step 6

Substitute a value from the interval $(- \infty, - 10)$ into the derivative to determine if the function is increasing or decreasing.

Replace the variable $x$ with $- 11$ in the expression.

$f' (- 11) = - \frac{- 11}{{(100 - {(- 11)}^{2})}^{\frac{1}{2}}}$

Simplify the result.

Simplify the denominator.

Simplify each term.

Raise $- 11$ to the power of $2$ .

$f' (- 11) = - \frac{- 11}{{(100 - 1 \cdot 121)}^{\frac{1}{2}}}$

Multiply $- 1$ by $121$ .

$f' (- 11) = - \frac{- 11}{{(100 - 121)}^{\frac{1}{2}}}$

Subtract $121$ from $100$ .

$f' (- 11) = - \frac{- 11}{{(- 21)}^{\frac{1}{2}}}$

Move the negative in front of the fraction.

$f' (- 11) = \frac{11}{{(- 21)}^{\frac{1}{2}}}$

The final answer is $\frac{11}{{(- 21)}^{\frac{1}{2}}}$ .

$\frac{11}{{(- 21)}^{\frac{1}{2}}}$

At $x = - 11$ the derivative is $\frac{11}{{(- 21)}^{\frac{1}{2}}}$ . Since this is negative, the function is decreasing on $(- \infty, - 10)$ .

Decreasing on $(- \infty, - 10)$ since $f' (x) < 0$

Step 7

Substitute a value from the interval $(- 10, 0)$ into the derivative to determine if the function is increasing or decreasing.

Replace the variable $x$ with $- 5$ in the expression.

$f' (- 5) = - \frac{- 5}{{(100 - {(- 5)}^{2})}^{\frac{1}{2}}}$

Simplify the result.

Simplify the denominator.

Simplify each term.

Raise $- 5$ to the power of $2$ .

$f' (- 5) = - \frac{- 5}{{(100 - 1 \cdot 25)}^{\frac{1}{2}}}$

Multiply $- 1$ by $25$ .

$f' (- 5) = - \frac{- 5}{{(100 - 25)}^{\frac{1}{2}}}$

Subtract $25$ from $100$ .

$f' (- 5) = - \frac{- 5}{75^{\frac{1}{2}}}$

Move the negative in front of the fraction.

$f' (- 5) = \frac{5}{75^{\frac{1}{2}}}$

The final answer is $\frac{5}{75^{\frac{1}{2}}}$ .

$\frac{5}{75^{\frac{1}{2}}}$

At $x = - 5$ the derivative is $\frac{5}{75^{\frac{1}{2}}}$ . Since this is positive, the function is increasing on $(- 10, 0)$ .

Increasing on $(- 10, 0)$ since $f' (x) > 0$

Step 8

Substitute a value from the interval $(0, 10)$ into the derivative to determine if the function is increasing or decreasing.

Replace the variable $x$ with $5$ in the expression.

$f' (5) = - \frac{5}{{(100 - {(5)}^{2})}^{\frac{1}{2}}}$

Simplify the result.

Simplify the denominator.

Simplify each term.

Raise $5$ to the power of $2$ .

$f' (5) = - \frac{5}{{(100 - 1 \cdot 25)}^{\frac{1}{2}}}$

Multiply $- 1$ by $25$ .

$f' (5) = - \frac{5}{{(100 - 25)}^{\frac{1}{2}}}$

Subtract $25$ from $100$ .

$f' (5) = - \frac{5}{75^{\frac{1}{2}}}$

The final answer is $- \frac{5}{75^{\frac{1}{2}}}$ .

$- \frac{5}{75^{\frac{1}{2}}}$

At $x = 5$ the derivative is $- \frac{5}{75^{\frac{1}{2}}}$ . Since this is negative, the function is decreasing on $(0, 10)$ .

Decreasing on $(0, 10)$ since $f' (x) < 0$

Step 9

Substitute a value from the interval $(10, \infty)$ into the derivative to determine if the function is increasing or decreasing.

Replace the variable $x$ with $11$ in the expression.

$f' (11) = - \frac{11}{{(100 - {(11)}^{2})}^{\frac{1}{2}}}$

Simplify the result.

Simplify the denominator.

Simplify each term.

Raise $11$ to the power of $2$ .

$f' (11) = - \frac{11}{{(100 - 1 \cdot 121)}^{\frac{1}{2}}}$

Multiply $- 1$ by $121$ .

$f' (11) = - \frac{11}{{(100 - 121)}^{\frac{1}{2}}}$

Subtract $121$ from $100$ .

$f' (11) = - \frac{11}{{(- 21)}^{\frac{1}{2}}}$

The final answer is $- \frac{11}{{(- 21)}^{\frac{1}{2}}}$ .

$- \frac{11}{{(- 21)}^{\frac{1}{2}}}$

At $x = 11$ the derivative is $- \frac{11}{{(- 21)}^{\frac{1}{2}}}$ . Since this is negative, the function is decreasing on $(10, \infty)$ .

Decreasing on $(10, \infty)$ since $f' (x) < 0$

Step 10

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

Increasing on: $(- 10, 0)$

Decreasing on: $(- \infty, - 10), (0, 10), (10, \infty)$

Step 11

Find Where Increasing/Decreasing Using Derivatives f(x) = square root of 100-x^2

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