Certainly! Let's perform logarithmic differentiation step-by-step to find the derivative of the given function:
[tex]\[ y = \frac{\sin x}{x^3 \ln x} \][/tex]
### Step 1: Taking the Natural Logarithm of Both Sides
First, we take the natural logarithm of both sides of the equation to help simplify the differentiation process:
[tex]\[ \ln y = \ln \left( \frac{\sin x}{x^3 \ln x} \right) \][/tex]
Using the properties of logarithms, we can split this into:
[tex]\[ \ln y = \ln (\sin x) - \ln (x^3 \ln x) \][/tex]
Next, using the properties of logarithms further:
[tex]\[ \ln y = \ln (\sin x) - \ln (x^3) - \ln (\ln x) \][/tex]
[tex]\[ \ln y = \ln (\sin x) - 3 \ln x - \ln (\ln x) \][/tex]
### Step 2: Differentiating Both Sides with Respect to [tex]\( x \)[/tex]
Now, we differentiate both sides with respect to [tex]\( x \)[/tex]. On the left side, we use the chain rule:
[tex]\[ \frac{d}{dx} (\ln y) = \frac{1}{y} \frac{dy}{dx} \][/tex]
On the right side, we differentiate each term separately:
[tex]\[ \frac{d}{dx} (\ln (\sin x)) = \frac{1}{\sin x} \cdot \cos x = \frac{\cos x}{\sin x} = \cot x \][/tex]
[tex]\[ \frac{d}{dx} (-3 \ln x) = -3 \cdot \frac{1}{x} = -\frac{3}{x} \][/tex]
[tex]\[ \frac{d}{dx} (-\ln (\ln x)) = -\frac{1}{\ln x} \cdot \frac{1}{x} = -\frac{1}{x \ln x} \][/tex]
Combining these results:
[tex]\[ \frac{1}{y} \frac{dy}{dx} = \cot x - \frac{3}{x} - \frac{1}{x \ln x} \][/tex]
### Step 3: Solving for [tex]\(\frac{dy}{dx}\)[/tex]
To find [tex]\(\frac{dy}{dx}\)[/tex], multiply both sides by [tex]\( y \)[/tex]:
[tex]\[ \frac{dy}{dx} = y \left( \cot x - \frac{3}{x} - \frac{1}{x \ln x} \right) \][/tex]
Now, recall the original function [tex]\( y = \frac{\sin x}{x^3 \ln x} \)[/tex]:
[tex]\[ \frac{dy}{dx} = \frac{\sin x}{x^3 \ln x} \left( \cot x - \frac{3}{x} - \frac{1}{x \ln x} \right) \][/tex]
### Step 4: Simplifying the Expression
To simplify the function:
[tex]\[ \frac{dy}{dx} = \frac{\sin x}{x^3 \ln x} \left( \frac{1}{\tan x} - \frac{3}{x} - \frac{1}{x \ln x} \right) \][/tex]
Further simplification:
[tex]\[ \frac{dy}{dx} = \left( \frac{1}{\tan x} - \frac{3}{x} - \frac{1}{x \ln x} \right) \frac{\sin x}{x^3 \ln x} \][/tex]
### Step 5: Final Simplified Form
So, the final derivative is:
[tex]\[ \frac{dy}{dx} = \left( \cot x - \frac{3}{x} - \frac{1}{x \ln x} \right) \frac{\sin x}{x^3 \ln x} \][/tex]
Simplifying further:
[tex]\[ \frac{dy}{dx} = \frac{x \ln x \cos x - 3 \ln x \sin x - \sin x}{x^4 (\ln x)^2} \][/tex]
Thus, the derivative of the given function [tex]\( y = \frac{\sin x}{x^3 \ln x} \)[/tex] using logarithmic differentiation is:
[tex]\[ \frac{dy}{dx} = \frac{x \ln x \cos x - 3 \ln x \sin x - \sin x}{x^4 (\ln x)^2} \][/tex]
