Calculus Examples

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

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 $1 - x^{\frac{3}{2}}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [- x^{\frac{3}{2}}]$ .

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

Step 1.1.1.2

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

$0 + \frac{d}{d x} [- x^{\frac{3}{2}}]$

Step 1.1.2

Evaluate $\frac{d}{d x} [- x^{\frac{3}{2}}]$ .

Tap for more steps...

Step 1.1.2.1

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

$0 - \frac{d}{d x} [x^{\frac{3}{2}}]$

Step 1.1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{3}{2}$ .

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

Step 1.1.2.3

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

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

Step 1.1.2.4

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

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

Step 1.1.2.5

Combine the numerators over the common denominator.

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

Step 1.1.2.6

Simplify the numerator.

Tap for more steps...

Step 1.1.2.6.1

Multiply $- 1$ by $2$ .

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

Step 1.1.2.6.2

Subtract $2$ from $3$ .

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

Step 1.1.2.7

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

$0 - \frac{3 x^{\frac{1}{2}}}{2}$

Step 1.1.3

Subtract $\frac{3 x^{\frac{1}{2}}}{2}$ from $0$ .

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

Step 1.2

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

$- \frac{3 x^{\frac{1}{2}}}{2}$

Step 2

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

Tap for more steps...

Step 2.1

Set the first derivative equal to $0$ .

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

Step 2.2

Set the numerator equal to zero.

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

Step 2.3

Solve the equation for $x$ .

Tap for more steps...

Step 2.3.1

Divide each term in $3 x^{\frac{1}{2}} = 0$ by $3$ and simplify.

Tap for more steps...

Step 2.3.1.1

Divide each term in $3 x^{\frac{1}{2}} = 0$ by $3$ .

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

Step 2.3.1.2

Simplify the left side.

Tap for more steps...

Step 2.3.1.2.1

Cancel the common factor.

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

Step 2.3.1.2.2

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

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

Step 2.3.1.3

Simplify the right side.

Tap for more steps...

Step 2.3.1.3.1

Divide $0$ by $3$ .

$x^{\frac{1}{2}} = 0$

Step 2.3.2

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

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

Step 2.3.3

Simplify the exponent.

Tap for more steps...

Step 2.3.3.1

Simplify the left side.

Tap for more steps...

Step 2.3.3.1.1

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

Tap for more steps...

Step 2.3.3.1.1.1

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

Tap for more steps...

Step 2.3.3.1.1.1.1

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

$x^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 2.3.3.1.1.1.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.3.3.1.1.1.2.1

Cancel the common factor.

$x^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 2.3.3.1.1.1.2.2

Rewrite the expression.

$x^{1} = 0^{2}$

Step 2.3.3.1.1.2

Simplify.

$x = 0^{2}$

Step 2.3.3.2

Simplify the right side.

Tap for more steps...

Step 2.3.3.2.1

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

$x = 0$

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.

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

Step 3.1.2

Anything raised to $1$ is the base itself.

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

Step 3.2

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

$x < 0$

Step 3.3

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 $0$ .

$x < 0$

$(- \infty, 0)$

$x < 0$

$(- \infty, 0)$

Step 4

Evaluate $1 - x^{\frac{3}{2}}$ at each $x$ value where the derivative is $0$ or undefined.

Tap for more steps...

Step 4.1

Evaluate at $x = 0$ .

Tap for more steps...

Step 4.1.1

Substitute $0$ for $x$ .

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

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

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

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

Step 4.1.2.1.2

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

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

Step 4.1.2.1.3

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.1.2.1.3.1

Cancel the common factor.

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

Step 4.1.2.1.3.2

Rewrite the expression.

$1 - 0^{3}$

Step 4.1.2.1.4

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

$1 - 0$

Step 4.1.2.1.5

Multiply $- 1$ by $0$ .

$1 + 0$

Step 4.1.2.2

Add $1$ and $0$ .

$1$

Step 4.2

List all of the points.

$(0, 1)$

Step 5