Calculus Examples

$f (x) = x + \sqrt{1 - x}$

Step 1

Find the first derivative.

Tap for more steps...

Step 1.1

Find the first derivative.

Tap for more steps...

Step 1.1.1

Differentiate.

Tap for more steps...

Step 1.1.1.1

By the Sum Rule, the derivative of $x + \sqrt{1 - x}$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [\sqrt{1 - x}]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [\sqrt{1 - x}]$

Step 1.1.1.2

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

$1 + \frac{d}{d x} [\sqrt{1 - x}]$

Step 1.1.2

Evaluate $\frac{d}{d x} [\sqrt{1 - x}]$ .

Tap for more steps...

Step 1.1.2.1

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

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

Step 1.1.2.2

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 $g (x) = 1 - x$ .

Tap for more steps...

Step 1.1.2.2.1

To apply the Chain Rule, set $u$ as $1 - x$ .

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

Step 1.1.2.2.2

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

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

Step 1.1.2.2.3

Replace all occurrences of $u$ with $1 - x$ .

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

Step 1.1.2.3

By the Sum Rule, the derivative of $1 - x$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [- x]$ .

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

Step 1.1.2.4

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

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

Step 1.1.2.5

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

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

Step 1.1.2.6

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

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

Step 1.1.2.7

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

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

Step 1.1.2.8

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

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

Step 1.1.2.9

Combine the numerators over the common denominator.

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

Step 1.1.2.10

Simplify the numerator.

Tap for more steps...

Step 1.1.2.10.1

Multiply $- 1$ by $2$ .

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

Step 1.1.2.10.2

Subtract $2$ from $1$ .

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

Step 1.1.2.11

Move the negative in front of the fraction.

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

Step 1.1.2.12

Multiply $- 1$ by $1$ .

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

Step 1.1.2.13

Subtract $1$ from $0$ .

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

Step 1.1.2.14

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

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

Step 1.1.2.15

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

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

Step 1.1.2.16

Move $- 1$ to the left of ${(1 - x)}^{- \frac{1}{2}}$ .

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

Step 1.1.2.17

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

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

Step 1.1.2.18

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

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

Step 1.1.2.19

Move the negative in front of the fraction.

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

Step 1.2

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

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

Step 2

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

Tap for more steps...

Step 2.1

Set the first derivative equal to $0$ .

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

Step 2.2

Subtract $1$ from both sides of the equation.

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

Step 2.3

Find the LCD of the terms in the equation.

Tap for more steps...

Step 2.3.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

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

Step 2.3.2

The LCM of one and any expression is the expression.

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

Step 2.4

Multiply each term in $- \frac{1}{2 {(1 - x)}^{\frac{1}{2}}} = - 1$ by $2 {(1 - x)}^{\frac{1}{2}}$ to eliminate the fractions.

Tap for more steps...

Step 2.4.1

Multiply each term in $- \frac{1}{2 {(1 - x)}^{\frac{1}{2}}} = - 1$ by $2 {(1 - x)}^{\frac{1}{2}}$ .

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

Step 2.4.2

Simplify the left side.

Tap for more steps...

Step 2.4.2.1

Cancel the common factor of $2 {(1 - x)}^{\frac{1}{2}}$ .

Tap for more steps...

Step 2.4.2.1.1

Move the leading negative in $- \frac{1}{2 {(1 - x)}^{\frac{1}{2}}}$ into the numerator.

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

Step 2.4.2.1.2

Cancel the common factor.

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

Step 2.4.2.1.3

Rewrite the expression.

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

Step 2.4.3

Simplify the right side.

Tap for more steps...

Step 2.4.3.1

Multiply $2$ by $- 1$ .

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

Step 2.5

Solve the equation.

Tap for more steps...

Step 2.5.1

Rewrite the equation as $- 2 {(1 - x)}^{\frac{1}{2}} = - 1$ .

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

Step 2.5.2

Divide each term in $- 2 {(1 - x)}^{\frac{1}{2}} = - 1$ by $- 2$ and simplify.

Tap for more steps...

Step 2.5.2.1

Divide each term in $- 2 {(1 - x)}^{\frac{1}{2}} = - 1$ by $- 2$ .

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

Step 2.5.2.2

Simplify the left side.

Tap for more steps...

Step 2.5.2.2.1

Cancel the common factor.

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

Step 2.5.2.2.2

Divide ${(1 - x)}^{\frac{1}{2}}$ by $1$ .

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

Step 2.5.2.3

Simplify the right side.

Tap for more steps...

Step 2.5.2.3.1

Dividing two negative values results in a positive value.

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

Step 2.5.3

Raise each side of the equation to the power of $2$ to eliminate the fractional exponent on the left side.

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

Step 2.5.4

Simplify the exponent.

Tap for more steps...

Step 2.5.4.1

Simplify the left side.

Tap for more steps...

Step 2.5.4.1.1

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

Tap for more steps...

Step 2.5.4.1.1.1

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

Tap for more steps...

Step 2.5.4.1.1.1.1

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

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

Step 2.5.4.1.1.1.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.5.4.1.1.1.2.1

Cancel the common factor.

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

Step 2.5.4.1.1.1.2.2

Rewrite the expression.

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

Step 2.5.4.1.1.2

Simplify.

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

Step 2.5.4.2

Simplify the right side.

Tap for more steps...

Step 2.5.4.2.1

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

Tap for more steps...

Step 2.5.4.2.1.1

Apply the product rule to $\frac{1}{2}$ .

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

Step 2.5.4.2.1.2

One to any power is one.

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

Step 2.5.4.2.1.3

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

$1 - x = \frac{1}{4}$

Step 2.5.5

Solve for $x$ .

Tap for more steps...

Step 2.5.5.1

Move all terms not containing $x$ to the right side of the equation.

Tap for more steps...

Step 2.5.5.1.1

Subtract $1$ from both sides of the equation.

$- x = \frac{1}{4} - 1$

Step 2.5.5.1.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$- x = \frac{1}{4} - 1 \cdot \frac{4}{4}$

Step 2.5.5.1.3

Combine $- 1$ and $\frac{4}{4}$ .

$- x = \frac{1}{4} + \frac{- 1 \cdot 4}{4}$

Step 2.5.5.1.4

Combine the numerators over the common denominator.

$- x = \frac{1 - 1 \cdot 4}{4}$

Step 2.5.5.1.5

Simplify the numerator.

Tap for more steps...

Step 2.5.5.1.5.1

Multiply $- 1$ by $4$ .

$- x = \frac{1 - 4}{4}$

Step 2.5.5.1.5.2

Subtract $4$ from $1$ .

$- x = \frac{- 3}{4}$

Step 2.5.5.1.6

Move the negative in front of the fraction.

$- x = - \frac{3}{4}$

Step 2.5.5.2

Divide each term in $- x = - \frac{3}{4}$ by $- 1$ and simplify.

Tap for more steps...

Step 2.5.5.2.1

Divide each term in $- x = - \frac{3}{4}$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- \frac{3}{4}}{- 1}$

Step 2.5.5.2.2

Simplify the left side.

Tap for more steps...

Step 2.5.5.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- \frac{3}{4}}{- 1}$

Step 2.5.5.2.2.2

Divide $x$ by $1$ .

$x = \frac{- \frac{3}{4}}{- 1}$

Step 2.5.5.2.3

Simplify the right side.

Tap for more steps...

Step 2.5.5.2.3.1

Dividing two negative values results in a positive value.

$x = \frac{\frac{3}{4}}{1}$

Step 2.5.5.2.3.2

Divide $\frac{3}{4}$ by $1$ .

$x = \frac{3}{4}$

Step 3

Find the values where the derivative is undefined.

Tap for more steps...

Step 3.1

Convert expressions with fractional exponents to radicals.

Tap for more steps...

Step 3.1.1

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

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

Step 3.1.2

Anything raised to $1$ is the base itself.

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

Step 3.2

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

$2 \sqrt{1 - x} = 0$

Step 3.3

Solve for $x$ .

Tap for more steps...

Step 3.3.1

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

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

Step 3.3.2

Simplify each side of the equation.

Tap for more steps...

Step 3.3.2.1

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

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

Step 3.3.2.2

Simplify the left side.

Tap for more steps...

Step 3.3.2.2.1

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

Tap for more steps...

Step 3.3.2.2.1.1

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

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

Step 3.3.2.2.1.2

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

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

Step 3.3.2.2.1.3

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

Tap for more steps...

Step 3.3.2.2.1.3.1

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

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

Step 3.3.2.2.1.3.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.3.2.2.1.3.2.1

Cancel the common factor.

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

Step 3.3.2.2.1.3.2.2

Rewrite the expression.

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

Step 3.3.2.2.1.4

Simplify.

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

Step 3.3.2.2.1.5

Apply the distributive property.

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

Step 3.3.2.2.1.6

Multiply.

Tap for more steps...

Step 3.3.2.2.1.6.1

Multiply $4$ by $1$ .

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

Step 3.3.2.2.1.6.2

Multiply $- 1$ by $4$ .

$4 - 4 x = 0^{2}$

Step 3.3.2.3

Simplify the right side.

Tap for more steps...

Step 3.3.2.3.1

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

$4 - 4 x = 0$

Step 3.3.3

Solve for $x$ .

Tap for more steps...

Step 3.3.3.1

Subtract $4$ from both sides of the equation.

$- 4 x = - 4$

Step 3.3.3.2

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

Tap for more steps...

Step 3.3.3.2.1

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

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

Step 3.3.3.2.2

Simplify the left side.

Tap for more steps...

Step 3.3.3.2.2.1

Cancel the common factor of $- 4$ .

Tap for more steps...

Step 3.3.3.2.2.1.1

Cancel the common factor.

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

Step 3.3.3.2.2.1.2

Divide $x$ by $1$ .

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

Step 3.3.3.2.3

Simplify the right side.

Tap for more steps...

Step 3.3.3.2.3.1

Divide $- 4$ by $- 4$ .

$x = 1$

Step 3.4

Set the radicand in $\sqrt{1 - x}$ less than $0$ to find where the expression is undefined.

$1 - x < 0$

Step 3.5

Solve for $x$ .

Tap for more steps...

Step 3.5.1

Subtract $1$ from both sides of the inequality.

$- x < - 1$

Step 3.5.2

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

Tap for more steps...

Step 3.5.2.1

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

$\frac{- x}{- 1} > \frac{- 1}{- 1}$

Step 3.5.2.2

Simplify the left side.

Tap for more steps...

Step 3.5.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} > \frac{- 1}{- 1}$

Step 3.5.2.2.2

Divide $x$ by $1$ .

$x > \frac{- 1}{- 1}$

Step 3.5.2.3

Simplify the right side.

Tap for more steps...

Step 3.5.2.3.1

Divide $- 1$ by $- 1$ .

$x > 1$

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 $0$ .

$x \geq 1$

$[1, \infty)$

$x \geq 1$

$[1, \infty)$

Step 4

Evaluate $x + \sqrt{1 - x}$ at each $x$ value where the derivative is $0$ or undefined.

Tap for more steps...

Step 4.1

Evaluate at $x = \frac{3}{4}$ .

Tap for more steps...

Step 4.1.1

Substitute $\frac{3}{4}$ for $x$ .

$(\frac{3}{4}) + \sqrt{1 - (\frac{3}{4})}$

Step 4.1.2

Simplify.

Tap for more steps...

Step 4.1.2.1

Simplify each term.

Tap for more steps...

Step 4.1.2.1.1

Write $1$ as a fraction with a common denominator.

$\frac{3}{4} + \sqrt{\frac{4}{4} - \frac{3}{4}}$

Step 4.1.2.1.2

Combine the numerators over the common denominator.

$\frac{3}{4} + \sqrt{\frac{4 - 3}{4}}$

Step 4.1.2.1.3

Subtract $3$ from $4$ .

$\frac{3}{4} + \sqrt{\frac{1}{4}}$

Step 4.1.2.1.4

Rewrite $\sqrt{\frac{1}{4}}$ as $\frac{\sqrt{1}}{\sqrt{4}}$ .

$\frac{3}{4} + \frac{\sqrt{1}}{\sqrt{4}}$

Step 4.1.2.1.5

Any root of $1$ is $1$ .

$\frac{3}{4} + \frac{1}{\sqrt{4}}$

Step 4.1.2.1.6

Simplify the denominator.

Tap for more steps...

Step 4.1.2.1.6.1

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

$\frac{3}{4} + \frac{1}{\sqrt{2^{2}}}$

Step 4.1.2.1.6.2

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

$\frac{3}{4} + \frac{1}{2}$

Step 4.1.2.2

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

$\frac{3}{4} + \frac{1}{2} \cdot \frac{2}{2}$

Step 4.1.2.3

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 4.1.2.3.1

Multiply $\frac{1}{2}$ by $\frac{2}{2}$ .

$\frac{3}{4} + \frac{2}{2 \cdot 2}$

Step 4.1.2.3.2

Multiply $2$ by $2$ .

$\frac{3}{4} + \frac{2}{4}$

Step 4.1.2.4

Combine the numerators over the common denominator.

$\frac{3 + 2}{4}$

Step 4.1.2.5

Add $3$ and $2$ .

$\frac{5}{4}$

Step 4.2

Evaluate at $x = 1$ .

Tap for more steps...

Step 4.2.1

Substitute $1$ for $x$ .

$(1) + \sqrt{1 - (1)}$

Step 4.2.2

Simplify.

Tap for more steps...

Step 4.2.2.1

Simplify each term.

Tap for more steps...

Step 4.2.2.1.1

Multiply $- 1$ by $1$ .

$1 + \sqrt{1 - 1}$

Step 4.2.2.1.2

Subtract $1$ from $1$ .

$1 + \sqrt{0}$

Step 4.2.2.1.3

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

$1 + \sqrt{0^{2}}$

Step 4.2.2.1.4

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

$1 + 0$

Step 4.2.2.2

Add $1$ and $0$ .

$1$

Step 4.3

List all of the points.

$(\frac{3}{4}, \frac{5}{4}), (1, 1)$

Step 5