Calculus Examples

$x^{2} y = 120$

Step 1

Divide each term in $x^{2} y = 120$ by $x^{2}$ and simplify.

Tap for more steps...

Step 1.1

Divide each term in $x^{2} y = 120$ by $x^{2}$ .

$\frac{x^{2} y}{x^{2}} = \frac{120}{x^{2}}$

Step 1.2

Simplify the left side.

Tap for more steps...

Step 1.2.1

Cancel the common factor of $x^{2}$ .

Tap for more steps...

Step 1.2.1.1

Cancel the common factor.

$\frac{x^{2} y}{x^{2}} = \frac{120}{x^{2}}$

Step 1.2.1.2

Divide $y$ by $1$ .

$y = \frac{120}{x^{2}}$

Step 2

Find the first derivative.

Tap for more steps...

Step 2.1

Find the first derivative.

Tap for more steps...

Step 2.1.1

Since $120$ is constant with respect to $x$ , the derivative of $\frac{120}{x^{2}}$ with respect to $x$ is $120 \frac{d}{d x} [\frac{1}{x^{2}}]$ .

$120 \frac{d}{d x} [\frac{1}{x^{2}}]$

Step 2.1.2

Apply basic rules of exponents.

Tap for more steps...

Step 2.1.2.1

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

$120 \frac{d}{d x} [{(x^{2})}^{- 1}]$

Step 2.1.2.2

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

Tap for more steps...

Step 2.1.2.2.1

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

$120 \frac{d}{d x} [x^{2 \cdot - 1}]$

Step 2.1.2.2.2

Multiply $2$ by $- 1$ .

$120 \frac{d}{d x} [x^{- 2}]$

Step 2.1.3

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

$120 (- 2 x^{- 3})$

Step 2.1.4

Multiply $- 2$ by $120$ .

$- 240 x^{- 3}$

Step 2.1.5

Simplify.

Tap for more steps...

Step 2.1.5.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- 240 \frac{1}{x^{3}}$

Step 2.1.5.2

Combine terms.

Tap for more steps...

Step 2.1.5.2.1

Combine $- 240$ and $\frac{1}{x^{3}}$ .

$\frac{- 240}{x^{3}}$

Step 2.1.5.2.2

Move the negative in front of the fraction.

$f' (x) = - \frac{240}{x^{3}}$

Step 2.2

The first derivative of $f (x)$ with respect to $x$ is $- \frac{240}{x^{3}}$ .

$- \frac{240}{x^{3}}$

Step 3

Set the first derivative equal to $0$ then solve the equation $- \frac{240}{x^{3}} = 0$ .

Tap for more steps...

Step 3.1

Set the first derivative equal to $0$ .

$- \frac{240}{x^{3}} = 0$

Step 3.2

Set the numerator equal to zero.

$240 = 0$

Step 3.3

Since $240 \neq 0$ , there are no solutions.

No solution

Step 4

Find the values where the derivative is undefined.

Tap for more steps...

Step 4.1

Set the denominator in $\frac{240}{x^{3}}$ equal to $0$ to find where the expression is undefined.

$x^{3} = 0$

Step 4.2

Solve for $x$ .

Tap for more steps...

Step 4.2.1

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

$x = \sqrt[3]{0}$

Step 4.2.2

Simplify $\sqrt[3]{0}$ .

Tap for more steps...

Step 4.2.2.1

Rewrite $0$ as $0^{3}$ .

$x = \sqrt[3]{0^{3}}$

Step 4.2.2.2

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

$x = 0$

Step 5

Evaluate $\frac{120}{x^{2}}$ at each $x$ value where the derivative is $0$ or undefined.

Tap for more steps...

Step 5.1

Evaluate at $x = 0$ .

Tap for more steps...

Step 5.1.1

Substitute $0$ for $x$ .

$\frac{120}{{(0)}^{2}}$

Step 5.1.2

Simplify.

Tap for more steps...

Step 5.1.2.1

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

$\frac{120}{0}$

Step 5.1.2.2

The expression contains a division by $0$ . The expression is undefined.

Undefined

Step 6

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

No critical points found