Logarithmic Differentiation 

Logarithmic differentiation is a technique used to differentiate complex functions, especially those involving products, quotients, or variables in exponents. By taking the natural logarithm of both sides of an equation, it simplifies differentiation using logarithmic identities. The steps involve applying \ln, simplifying, differentiating implicitly, and solving for the derivative. It's especially useful for functions like $y=x^x$ or $y=\frac{(x^2+1{)}^3}{\sqrt{x-1}}$ . This method reduces complicated expressions into simpler, more manageable forms for differentiation.

1.0What is Logarithmic Differentiation?

Logarithmic Differentiation is a technique where we take the natural logarithm (usually log base e) on both sides of a function and then differentiate implicitly. It’s particularly effective for:

• Products of multiple functions
• Quotients
• Exponential functions like $f(x)=(x^2+1{)}^x$

2.0Logarithmic Differentiation Formula

If y = f(x), then using logarithmic differentiation:

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

Differentiate both sides:

$\frac{1}{y}\cdot \frac{dy}{dx}=\frac{d}{dx}[\ln f(x)]$

Then,

$\frac{dy}{dx}=y\cdot \frac{d}{dx}[\ln f(x)]=f(x)\cdot \frac{d}{dx}[\ln f(x)]$

3.0Solved Examples of Logarithmic Differentiation

Example 1: Find the derivative of $y=x^x$

Solution:

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

Differentiate both sides:

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

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

Example 2: Differentiate $y=\frac{(x^2+3{)}^4\cdot \sqrt{x+1}}{(2x^2+1{)}^3}$

Solution:

Take natural log on both sides:

$\ln y=\ln \left[\frac{(x^2+3{)}^4\cdot (x+1{)}^{1/2}}{(2x^2+1{)}^3}\right]$

Apply log laws:

$\ln y=4\ln (x^2+3)+\frac{1}{2}\ln (x+1)-3\ln (2x^2+1)$

Differentiate:

$$\begin{gathered} \frac{1}{y}\frac{dy}{dx}=\frac{4\cdot 2x}{x^2+3}+\frac{1}{2(x+1)}-\frac{3\cdot 4x}{2x^2+1} \\ \frac{dy}{dx}=y\left[\frac{8x}{x^2+3}+\frac{1}{2(x+1)}-\frac{12x}{2x^2+1}\right] \end{gathered}$$

Substitute original y back:

$\frac{dy}{dx}=\frac{(x^2+3{)}^4\cdot \sqrt{x+1}}{(2x^2+1{)}^3}\left[\frac{8x}{x^2+3}+\frac{1}{2(x+1)}-\frac{12x}{2x^2+1}\right]$

Example 3: Differentiate $y=\frac{(x^2+1{)}^3}{\sqrt{x-1}}$

Solution: 

Take natural log:

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

Differentiate both sides:

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

Multiply by yy:

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

Example 4: Find the derivative of y wrt x. $y=(\sin x{)}^{\tan x}$

Solution:

Take logarithm:

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

Differentiate:

$\frac{1}{y}\frac{dy}{dx}={\sec }^{2}x\cdot \ln (\sin x)+\tan x\cdot \frac{\cos x}{\sin x}={\sec }^{2}x\cdot \ln (\sin x)+\tan x\cdot \cot x={\sec }^{2}x\cdot \ln (\sin x)+1$

So,

$\frac{dy}{dx}=(\sin x{)}^{\tan x}({\sec }^{2}x\cdot \ln (\sin x)+1)$

Example 5: Differentiate $y=(x^2+3x+2{)}^{\sqrt{x^2+1}}$

Solution:

Step 1: Take natural log:

$\ln y=\sqrt{x^2+1}\cdot \ln (x^2+3x+2)$

Step 2: Differentiate:

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

Final Answer:

$\frac{dy}{dx}=(x^2+3x+2{)}^{\sqrt{x^2+1}}\left[\frac{x\ln (x^2+3x+2)}{\sqrt{x^2+1}}+\sqrt{x^2+1}\cdot \frac{2x+3}{x^2+3x+2}\right]$

Example 6: Differentiate $y=\frac{[\ln x{]}^x}{x^{\ln x}}$

Solution:

Take natural log:

$\ln y=x\ln (\ln x)-\ln x\cdot \ln x=x\ln (\ln x)-(\ln x{)}^2$

Differentiate:

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

Simplify:

$\frac{dy}{dx}=\frac{[\ln x{]}^x}{x^{\ln x}}\left[\ln (\ln x)+\frac{1}{\ln x}-\frac{2\ln x}{x}\right]$

Example 7: Differentiate $y={\left(\frac{(x^2+1{)}^5\cdot \sqrt{\tan x}}{e^x\cdot \ln x}\right)}^x$

Solution:

Take logarithm:

$\ln y=x\cdot \ln \left(\frac{(x^2+1{)}^5\cdot \sqrt{\tan x}}{e^x\cdot \ln x}\right)$

Break it down using log rules:

$\ln y=x\left[5\ln (x^2+1)+\frac{1}{2}\ln (\tan x)-x-\ln (\ln x)\right]$

Differentiate using product rule:

$\frac{1}{y}\frac{dy}{dx}=\left[5\ln (x^2+1)+\frac{1}{2}\ln (\tan x)-x-\ln (\ln x)\right]+x\left[\frac{10x}{x^2+1}+\frac{1}{2}\frac{1}{\tan x}{\sec }^{2}x-1-\frac{1}{\ln x}\cdot \frac{1}{x}\right]$

You can now write:

$\frac{dy}{dx}=y\cdot (\text{expression above})$

4.0Logarithmic Differentiation Practice Questions

1. Differentiate: $y=\frac{(3x^2+5{)}^2}{\sqrt{x^2+1}}$
2. Differentiate: $y=x^x\cdot \sqrt{x+2}$
3. Differentiate: $y=\frac{(x^2+1{)}^3\cdot (x+1{)}^2}{\sqrt{2x^3+4}}$
4. Find if $\frac{dy}{dx}\text{ if }y={\left(\frac{x^2+3}{x^2-1}\right)}^x$
5. Let $y=\frac{(x^3+1{)}^5\cdot (x^2-4{)}^3}{(x^2+2x+2{)}^4\cdot \sqrt{x-2}}$_. Find using logarithmic differentiation.
6. Differentiate: $\frac{dy}{dx}$
7. Differentiate: $y=\frac{x^3\sqrt{x^2+1}}{(x-1{)}^2}$
8. Differentiate: $y=(x^2+4{)}^x\cdot e^{x\ln x}$

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