Logarithmic Differentiation

What Is Logarithmic Differentiation?

Logarithmic differentiation is a method that uses the natural logarithm to simplify the differentiation of complicated functions.

You take the natural log of both sides of an equation $$y = f(x)$$,

then use logarithm properties and the chain rule to find $$\frac{dy}{dx}$$​.

Why use Logatithmic Differentiation?

It’s especially useful when:

• y is a product or quotient of many functions.

• y is a function raised to another function

Example:

$$y=(x^{2}+5)^{6}(3x-7)^{7} , y=(\sin{x})^{\cos{x}}$$

using logarithmic differentiation, it becomes much easier.

General Steps

1. Take the natural log of both sides:

   $$ln(y) = ln(f(x))$$

2. Simplify using log rules:

   • $$ln(ab) = ln(a) + ln(b)$$

   • $$ln\left(\frac{a}{b}\right) = ln(a) – ln(b)$$

   • $$ln(a^{n}) = n \cdot ln(a)$$

3. Differentiate both sides with respect to x:
   Use the chain rule on the left:

   $$\frac{1}{y}\frac{dy}{dx} = \frac{f^\prime(x)}{f(x)}$$

4. Multiply both sides by $y$:

   $$\frac{dy}{dx} = y \cdot \frac{f^\prime(x)}{f(x)}$$

5. Substitute $$y=f(x)$$.

Example 1

$$y = x^{x}$$

Step 1: Take $$ln$$ of both sides:

$$ln(y) = ln(x^{x})$$

Step 2: Simplify using log rules:

$$ln(y) = x \cdot ln(x)$$

Step 3: Differentiate both sides:

$$\frac{1}{y}\frac{dy}{dx} = ln(x) + x \cdot \frac{1}{x} = ln(x) + 1$$

Step 4: Multiply both sides by y:

$$\frac{dy}{dx} =y \cdot (ln(x) + 1)$$

Step 5: Substitute $$y = x^{x}$$:

$$\frac{dy}{dx} = x^{x}(ln(x) + 1)$$​

Example 2

$$y = (x^{2}+1)^{3}(3x-2)^{5}$$

Step 1 Take $$ln$$ of both sides:

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

Step 2: Differentiate both sides:

$$\frac{1}{y}\frac{dy}{dx} = 3 \cdot \frac{2x}{x^{2}+1} + 5 \cdot \frac{3}{3x-2}$$​

Step 3: Multiply both sides by $$y=(x^{2}+1)^{3}(3x-2)^{5}$$:

$$\frac{dy}{dx} =y \cdot  (3 \cdot \frac{2x}{x^{2}+1} + 5 \cdot \frac{3}{3x-2})$$

$$= (x^{2}+1)^{3}(3x-2)^{5} \left(\frac{6x}{x^{2}+1} + \frac{15}{3x-2}\right)$$​​

Example 3 (Quotient & Exponent)

$$f(x) = \frac{(x^2+1)^3}{(2x-5)^2}$$

​Step 1 Take $$ln$$ of both sides:

$$ln(y) = ln\left(\frac{(x^2+1)^3}{(2x-5)^2}\right) = 3\cdot ln(x^{2}+1)-2\cdot (2x-5)$$

Step 2: Differentiate both sides:

$$\frac{1}{y}\frac{dy}{dx} = 3\cdot\frac{2x}{x^{2}+1}-2\cdot\frac{2}{2x-5}$$

Step 3: Multiply both sides by $$y$$:

$$\frac{dy}{dx} = y \cdot \left(\frac{6x}{x^{2}+1}-\frac{4}{2x-5}\right)$$

Step 4: Substitute $$y = \frac{(x^2+1)^3}{(2x-5)^2}$$:

$$\frac{dy}{dx} = \frac{(x^2+1)^3}{(2x-5)^2} \cdot \left(\frac{6x}{x^{2}+1}-\frac{4}{2x-5}\right)$$