Calculus Examples

$x \ln (3 y) = 5 x y^{3}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (x \ln (3 y)) = \frac{d}{d x} (5 x y^{3})$

Step 2

Differentiate the left side of the equation.

Tap for more steps...

Step 2.1

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 $g (x) = \ln (3 y)$ .

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

Step 2.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \ln (x)$ and $g (x) = 3 y$ .

Tap for more steps...

Step 2.2.1

To apply the Chain Rule, set $u$ as $3 y$ .

$x (\frac{d}{d u} [\ln (u)] \frac{d}{d x} [3 y]) + \ln (3 y) \frac{d}{d x} [x]$

Step 2.2.2

The derivative of $\ln (u)$ with respect to $u$ is $\frac{1}{u}$ .

$x (\frac{1}{u} \frac{d}{d x} [3 y]) + \ln (3 y) \frac{d}{d x} [x]$

Step 2.2.3

Replace all occurrences of $u$ with $3 y$ .

$x (\frac{1}{3 y} \frac{d}{d x} [3 y]) + \ln (3 y) \frac{d}{d x} [x]$

Step 2.3

Differentiate using the Constant Multiple Rule.

Tap for more steps...

Step 2.3.1

Combine $\frac{1}{3 y}$ and $x$ .

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

Step 2.3.2

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

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

Step 2.3.3

Simplify terms.

Tap for more steps...

Step 2.3.3.1

Combine $3$ and $\frac{x}{3 y}$ .

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

Step 2.3.3.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 2.3.3.2.1

Cancel the common factor.

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

Step 2.3.3.2.2

Rewrite the expression.

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

Step 2.4

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

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

Step 2.5

Combine $\frac{x}{y}$ and $y'$ .

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

Step 2.6

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

$\frac{x y'}{y} + \ln (3 y) \cdot 1$

Step 2.7

Multiply $\ln (3 y)$ by $1$ .

$\frac{x y'}{y} + \ln (3 y)$

Step 3

Differentiate the right side of the equation.

Tap for more steps...

Step 3.1

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

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

Step 3.2

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 $g (x) = y^{3}$ .

$5 (x \frac{d}{d x} [y^{3}] + y^{3} \frac{d}{d x} [x])$

Step 3.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = y$ .

Tap for more steps...

Step 3.3.1

To apply the Chain Rule, set $u$ as $y$ .

$5 (x (\frac{d}{d u} [u^{3}] \frac{d}{d x} [y]) + y^{3} \frac{d}{d x} [x])$

Step 3.3.2

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

$5 (x (3 u^{2} \frac{d}{d x} [y]) + y^{3} \frac{d}{d x} [x])$

Step 3.3.3

Replace all occurrences of $u$ with $y$ .

$5 (x (3 y^{2} \frac{d}{d x} [y]) + y^{3} \frac{d}{d x} [x])$

Step 3.4

Move $3$ to the left of $x$ .

$5 (3 \cdot x y^{2} \frac{d}{d x} [y] + y^{3} \frac{d}{d x} [x])$

Step 3.5

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

$5 (3 x y^{2} y' + y^{3} \frac{d}{d x} [x])$

Step 3.6

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

$5 (3 x y^{2} y' + y^{3} \cdot 1)$

Step 3.7

Multiply $y^{3}$ by $1$ .

$5 (3 x y^{2} y' + y^{3})$

Step 3.8

Simplify.

Tap for more steps...

Step 3.8.1

Apply the distributive property.

$5 (3 x y^{2} y') + 5 y^{3}$

Step 3.8.2

Multiply $3$ by $5$ .

$15 x y^{2} y' + 5 y^{3}$

Step 3.8.3

Reorder terms.

$5 y^{3} + 15 y^{2} x y'$

Step 4

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

$\frac{x y'}{y} + \ln (3 y) = 5 y^{3} + 15 y^{2} x y'$

Step 5

Solve for $y'$ .

Tap for more steps...

Step 5.1

Subtract $15 y^{2} x y'$ from both sides of the equation.

$\frac{x y'}{y} + \ln (3 y) - 15 y^{2} x y' = 5 y^{3}$

Step 5.2

Find the LCD of the terms in the equation.

Tap for more steps...

Step 5.2.1

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

$y, 1, 1, 1$

Step 5.2.2

The LCM of one and any expression is the expression.

$y$

Step 5.3

Multiply each term in $\frac{x y'}{y} + \ln (3 y) - 15 y^{2} x y' = 5 y^{3}$ by $y$ to eliminate the fractions.

Tap for more steps...

Step 5.3.1

Multiply each term in $\frac{x y'}{y} + \ln (3 y) - 15 y^{2} x y' = 5 y^{3}$ by $y$ .

$\frac{x y'}{y} y + \ln (3 y) y - 15 y^{2} x y' y = 5 y^{3} y$

Step 5.3.2

Simplify the left side.

Tap for more steps...

Step 5.3.2.1

Simplify each term.

Tap for more steps...

Step 5.3.2.1.1

Cancel the common factor of $y$ .

Tap for more steps...

Step 5.3.2.1.1.1

Cancel the common factor.

$\frac{x y'}{y} y + \ln (3 y) y - 15 y^{2} x y' y = 5 y^{3} y$

Step 5.3.2.1.1.2

Rewrite the expression.

$x y' + \ln (3 y) y - 15 y^{2} x y' y = 5 y^{3} y$

Step 5.3.2.1.2

Multiply $y^{2}$ by $y$ by adding the exponents.

Tap for more steps...

Step 5.3.2.1.2.1

Move $y$ .

$x y' + \ln (3 y) y - 15 (y \cdot y^{2}) x y' = 5 y^{3} y$

Step 5.3.2.1.2.2

Multiply $y$ by $y^{2}$ .

Tap for more steps...

Step 5.3.2.1.2.2.1

Raise $y$ to the power of $1$ .

$x y' + \ln (3 y) y - 15 (y^{1} y^{2}) x y' = 5 y^{3} y$

Step 5.3.2.1.2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x y' + \ln (3 y) y - 15 y^{1 + 2} x y' = 5 y^{3} y$

Step 5.3.2.1.2.3

Add $1$ and $2$ .

$x y' + \ln (3 y) y - 15 y^{3} x y' = 5 y^{3} y$

Step 5.3.2.2

Reorder factors in $x y' + \ln (3 y) y - 15 y^{3} x y'$ .

$x y' + y \ln (3 y) - 15 y^{3} x y' = 5 y^{3} y$

Step 5.3.3

Simplify the right side.

Tap for more steps...

Step 5.3.3.1

Multiply $y^{3}$ by $y$ by adding the exponents.

Tap for more steps...

Step 5.3.3.1.1

Move $y$ .

$x y' + y \ln (3 y) - 15 y^{3} x y' = 5 (y \cdot y^{3})$

Step 5.3.3.1.2

Multiply $y$ by $y^{3}$ .

Tap for more steps...

Step 5.3.3.1.2.1

Raise $y$ to the power of $1$ .

$x y' + y \ln (3 y) - 15 y^{3} x y' = 5 (y^{1} y^{3})$

Step 5.3.3.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x y' + y \ln (3 y) - 15 y^{3} x y' = 5 y^{1 + 3}$

Step 5.3.3.1.3

Add $1$ and $3$ .

$x y' + y \ln (3 y) - 15 y^{3} x y' = 5 y^{4}$

Step 5.4

Solve the equation.

Tap for more steps...

Step 5.4.1

Subtract $y \ln (3 y)$ from both sides of the equation.

$x y' - 15 y^{3} x y' = 5 y^{4} - y \ln (3 y)$

Step 5.4.2

Factor $x y'$ out of $x y' - 15 y^{3} x y'$ .

Tap for more steps...

Step 5.4.2.1

Factor $x y'$ out of $x y'$ .

$x y' \cdot 1 - 15 y^{3} x y' = 5 y^{4} - y \ln (3 y)$

Step 5.4.2.2

Factor $x y'$ out of $- 15 y^{3} x y'$ .

$x y' \cdot 1 + x y' (- 15 y^{3}) = 5 y^{4} - y \ln (3 y)$

Step 5.4.2.3

Factor $x y'$ out of $x y' \cdot 1 + x y' (- 15 y^{3})$ .

$x y' (1 - 15 y^{3}) = 5 y^{4} - y \ln (3 y)$

Step 5.4.3

Divide each term in $x y' (1 - 15 y^{3}) = 5 y^{4} - y \ln (3 y)$ by $x (1 - 15 y^{3})$ and simplify.

Tap for more steps...

Step 5.4.3.1

Divide each term in $x y' (1 - 15 y^{3}) = 5 y^{4} - y \ln (3 y)$ by $x (1 - 15 y^{3})$ .

$\frac{x y' (1 - 15 y^{3})}{x (1 - 15 y^{3})} = \frac{5 y^{4}}{x (1 - 15 y^{3})} + \frac{- y \ln (3 y)}{x (1 - 15 y^{3})}$

Step 5.4.3.2

Simplify the left side.

Tap for more steps...

Step 5.4.3.2.1

Cancel the common factor of $x$ .

Tap for more steps...

Step 5.4.3.2.1.1

Cancel the common factor.

$\frac{x y' (1 - 15 y^{3})}{x (1 - 15 y^{3})} = \frac{5 y^{4}}{x (1 - 15 y^{3})} + \frac{- y \ln (3 y)}{x (1 - 15 y^{3})}$

Step 5.4.3.2.1.2

Rewrite the expression.

$\frac{y' (1 - 15 y^{3})}{1 - 15 y^{3}} = \frac{5 y^{4}}{x (1 - 15 y^{3})} + \frac{- y \ln (3 y)}{x (1 - 15 y^{3})}$

Step 5.4.3.2.2

Cancel the common factor of $1 - 15 y^{3}$ .

Tap for more steps...

Step 5.4.3.2.2.1

Cancel the common factor.

$\frac{y' (1 - 15 y^{3})}{1 - 15 y^{3}} = \frac{5 y^{4}}{x (1 - 15 y^{3})} + \frac{- y \ln (3 y)}{x (1 - 15 y^{3})}$

Step 5.4.3.2.2.2

Divide $y'$ by $1$ .

$y' = \frac{5 y^{4}}{x (1 - 15 y^{3})} + \frac{- y \ln (3 y)}{x (1 - 15 y^{3})}$

Step 5.4.3.3

Simplify the right side.

Tap for more steps...

Step 5.4.3.3.1

Move the negative in front of the fraction.

$y' = \frac{5 y^{4}}{x (1 - 15 y^{3})} - \frac{y \ln (3 y)}{x (1 - 15 y^{3})}$

Step 6

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

$\frac{d y}{d x} = \frac{5 y^{4}}{x (1 - 15 y^{3})} - \frac{y \ln (3 y)}{x (1 - 15 y^{3})}$