Calculus Examples

$5 x^{3} = - 3 x y + 2$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (5 x^{3}) = \frac{d}{d x} (- 3 x y + 2)$

Step 2

Differentiate the left side of the equation.

Tap for more steps...

Step 2.1

Since $5$ is constant with respect to $x$ , the derivative of $5 x^{3}$ with respect to $x$ is $5 \frac{d}{d x} [x^{3}]$ .

$5 \frac{d}{d x} [x^{3}]$

Step 2.2

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

$5 (3 x^{2})$

Step 2.3

Multiply $3$ by $5$ .

$15 x^{2}$

Step 3

Differentiate the right side of the equation.

Tap for more steps...

Step 3.1

By the Sum Rule, the derivative of $- 3 x y + 2$ with respect to $x$ is $\frac{d}{d x} [- 3 x y] + \frac{d}{d x} [2]$ .

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

Step 3.2

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

Tap for more steps...

Step 3.2.1

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

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

Step 3.2.2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = y$ .

$- 3 (x \frac{d}{d x} [y] + y \frac{d}{d x} [x]) + \frac{d}{d x} [2]$

Step 3.2.3

Rewrite $\frac{d}{d x} [y]$ as $y'$ .

$- 3 (x y' + y \frac{d}{d x} [x]) + \frac{d}{d x} [2]$

Step 3.2.4

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

$- 3 (x y' + y \cdot 1) + \frac{d}{d x} [2]$

Step 3.2.5

Multiply $y$ by $1$ .

$- 3 (x y' + y) + \frac{d}{d x} [2]$

Step 3.3

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$- 3 (x y' + y) + 0$

Step 3.4

Simplify.

Tap for more steps...

Step 3.4.1

Apply the distributive property.

$- 3 (x y') - 3 y + 0$

Step 3.4.2

Add $- 3 x y' - 3 y$ and $0$ .

$- 3 x y' - 3 y$

Step 4

Reform the equation by setting the left side equal to the right side.

$15 x^{2} = - 3 x y' - 3 y$

Step 5

Solve for $y'$ .

Tap for more steps...

Step 5.1

Rewrite the equation as $- 3 x y' - 3 y = 15 x^{2}$ .

$- 3 x y' - 3 y = 15 x^{2}$

Step 5.2

Add $3 y$ to both sides of the equation.

$- 3 x y' = 15 x^{2} + 3 y$

Step 5.3

Divide each term in $- 3 x y' = 15 x^{2} + 3 y$ by $- 3 x$ and simplify.

Tap for more steps...

Step 5.3.1

Divide each term in $- 3 x y' = 15 x^{2} + 3 y$ by $- 3 x$ .

$\frac{- 3 x y'}{- 3 x} = \frac{15 x^{2}}{- 3 x} + \frac{3 y}{- 3 x}$

Step 5.3.2

Simplify the left side.

Tap for more steps...

Step 5.3.2.1

Cancel the common factor of $- 3$ .

Tap for more steps...

Step 5.3.2.1.1

Cancel the common factor.

$\frac{- 3 x y'}{- 3 x} = \frac{15 x^{2}}{- 3 x} + \frac{3 y}{- 3 x}$

Step 5.3.2.1.2

Rewrite the expression.

$\frac{x y'}{x} = \frac{15 x^{2}}{- 3 x} + \frac{3 y}{- 3 x}$

Step 5.3.2.2

Cancel the common factor of $x$ .

Tap for more steps...

Step 5.3.2.2.1

Cancel the common factor.

$\frac{x y'}{x} = \frac{15 x^{2}}{- 3 x} + \frac{3 y}{- 3 x}$

Step 5.3.2.2.2

Divide $y'$ by $1$ .

$y' = \frac{15 x^{2}}{- 3 x} + \frac{3 y}{- 3 x}$

Step 5.3.3

Simplify the right side.

Tap for more steps...

Step 5.3.3.1

Simplify each term.

Tap for more steps...

Step 5.3.3.1.1

Cancel the common factor of $15$ and $- 3$ .

Tap for more steps...

Step 5.3.3.1.1.1

Factor $3$ out of $15 x^{2}$ .

$y' = \frac{3 (5 x^{2})}{- 3 x} + \frac{3 y}{- 3 x}$

Step 5.3.3.1.1.2

Cancel the common factors.

Tap for more steps...

Step 5.3.3.1.1.2.1

Factor $3$ out of $- 3 x$ .

$y' = \frac{3 (5 x^{2})}{3 (- x)} + \frac{3 y}{- 3 x}$

Step 5.3.3.1.1.2.2

Cancel the common factor.

$y' = \frac{3 (5 x^{2})}{3 (- x)} + \frac{3 y}{- 3 x}$

Step 5.3.3.1.1.2.3

Rewrite the expression.

$y' = \frac{5 x^{2}}{- x} + \frac{3 y}{- 3 x}$

Step 5.3.3.1.2

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

Tap for more steps...

Step 5.3.3.1.2.1

Factor $x$ out of $5 x^{2}$ .

$y' = \frac{x (5 x)}{- x} + \frac{3 y}{- 3 x}$

Step 5.3.3.1.2.2

Move the negative one from the denominator of $\frac{5 x}{- 1}$ .

$y' = - 1 \cdot (5 x) + \frac{3 y}{- 3 x}$

Step 5.3.3.1.3

Rewrite $- 1 \cdot (5 x)$ as $- (5 x)$ .

$y' = - (5 x) + \frac{3 y}{- 3 x}$

Step 5.3.3.1.4

Multiply $5$ by $- 1$ .

$y' = - 5 x + \frac{3 y}{- 3 x}$

Step 5.3.3.1.5

Cancel the common factor of $3$ and $- 3$ .

Tap for more steps...

Step 5.3.3.1.5.1

Factor $3$ out of $3 y$ .

$y' = - 5 x + \frac{3 (y)}{- 3 x}$

Step 5.3.3.1.5.2

Cancel the common factors.

Tap for more steps...

Step 5.3.3.1.5.2.1

Factor $3$ out of $- 3 x$ .

$y' = - 5 x + \frac{3 (y)}{3 (- x)}$

Step 5.3.3.1.5.2.2

Cancel the common factor.

$y' = - 5 x + \frac{3 y}{3 (- x)}$

Step 5.3.3.1.5.2.3

Rewrite the expression.

$y' = - 5 x + \frac{y}{- x}$

Step 5.3.3.1.6

Move the negative in front of the fraction.

$y' = - 5 x - \frac{y}{x}$

Step 6

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = - 5 x - \frac{y}{x}$

Step 7

Set $\frac{d y}{d x} = 0$ then solve for $x$ in terms of $y$ .

Tap for more steps...

Step 7.1

Find the LCD of the terms in the equation.

Tap for more steps...

Step 7.1.1

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

$1, x, 1$

Step 7.1.2

The LCM of one and any expression is the expression.

$x$

Step 7.2

Multiply each term in $- 5 x - \frac{y}{x} = 0$ by $x$ to eliminate the fractions.

Tap for more steps...

Step 7.2.1

Multiply each term in $- 5 x - \frac{y}{x} = 0$ by $x$ .

$- 5 x \cdot x - \frac{y}{x} x = 0 x$

Step 7.2.2

Simplify the left side.

Tap for more steps...

Step 7.2.2.1

Simplify each term.

Tap for more steps...

Step 7.2.2.1.1

Multiply $x$ by $x$ by adding the exponents.

Tap for more steps...

Step 7.2.2.1.1.1

Move $x$ .

$- 5 (x \cdot x) - \frac{y}{x} x = 0 x$

Step 7.2.2.1.1.2

Multiply $x$ by $x$ .

$- 5 x^{2} - \frac{y}{x} x = 0 x$

Step 7.2.2.1.2

Cancel the common factor of $x$ .

Tap for more steps...

Step 7.2.2.1.2.1

Move the leading negative in $- \frac{y}{x}$ into the numerator.

$- 5 x^{2} + \frac{- y}{x} x = 0 x$

Step 7.2.2.1.2.2

Cancel the common factor.

$- 5 x^{2} + \frac{- y}{x} x = 0 x$

Step 7.2.2.1.2.3

Rewrite the expression.

$- 5 x^{2} - y = 0 x$

Step 7.2.3

Simplify the right side.

Tap for more steps...

Step 7.2.3.1

Multiply $0$ by $x$ .

$- 5 x^{2} - y = 0$

Step 7.3

Solve the equation.

Tap for more steps...

Step 7.3.1

Add $y$ to both sides of the equation.

$- 5 x^{2} = y$

Step 7.3.2

Divide each term in $- 5 x^{2} = y$ by $- 5$ and simplify.

Tap for more steps...

Step 7.3.2.1

Divide each term in $- 5 x^{2} = y$ by $- 5$ .

$\frac{- 5 x^{2}}{- 5} = \frac{y}{- 5}$

Step 7.3.2.2

Simplify the left side.

Tap for more steps...

Step 7.3.2.2.1

Cancel the common factor of $- 5$ .

Tap for more steps...

Step 7.3.2.2.1.1

Cancel the common factor.

$\frac{- 5 x^{2}}{- 5} = \frac{y}{- 5}$

Step 7.3.2.2.1.2

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

$x^{2} = \frac{y}{- 5}$

Step 7.3.2.3

Simplify the right side.

Tap for more steps...

Step 7.3.2.3.1

Move the negative in front of the fraction.

$x^{2} = - \frac{y}{5}$

Step 7.3.3

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

$x = \pm \sqrt{- \frac{y}{5}}$

Step 7.3.4

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

Tap for more steps...

Step 7.3.4.1

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

$x = \sqrt{- \frac{y}{5}}$

Step 7.3.4.2

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

$x = - \sqrt{- \frac{y}{5}}$

Step 7.3.4.3

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

$x = \sqrt{- \frac{y}{5}}$

$x = - \sqrt{- \frac{y}{5}}$

$x = \sqrt{- \frac{y}{5}}$

$x = - \sqrt{- \frac{y}{5}}$

$x = \sqrt{- \frac{y}{5}}$

$x = - \sqrt{- \frac{y}{5}}$

$x = \sqrt{- \frac{y}{5}}$

$x = - \sqrt{- \frac{y}{5}}$

Step 8

Calculated $x$ values cannot contain imaginary components.

$\sqrt{- \frac{y}{5}}$ is not a valid value for x

Step 9

Calculated $x$ values cannot contain imaginary components.

$- \sqrt{- \frac{y}{5}}$ is not a valid value for x

Step 10

No points that set $\frac{d y}{d x} = 0$ are on the real number plane.

No Points

Step 11