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To determine \( y' \) from the given equation:

\[
\ln(10) \cdot \log\left(\frac{x}{y}\right) + \cos(xy) = 5
\]

we will follow these steps:

1. **Implicit Differentiation**: Since \( y \) is a function of \( x \), we differentiate both sides of the equation with respect to \( x \), treating \( y \) as a function of \( x \).

2. **Apply Chain Rule**: Wherever \( y \) appears, apply the chain rule.

Let’s begin the differentiation.

### Step 1: Differentiate both sides with respect to \( x \):

Differentiating the left side term-by-term:

- First term: \( \ln(10) \cdot \log\left(\frac{x}{y}\right) \)

  \[
  \frac{d}{dx}\left[\ln(10) \cdot \log\left(\frac{x}{y}\right)\right]
  \]
  
  Using the product rule:
  
  \[
  \ln(10) \cdot \frac{d}{dx}\left[\log\left(\frac{x}{y}\right)\right]
  \]

  Now, for the derivative of \( \log\left(\frac{x}{y}\right) \):
  
  \[
  \frac{d}{dx}\left[\log\left(\frac{x}{y}\right)\right] = \frac{1}{\frac{x}{y}} \cdot \frac{d}{dx}\left[\frac{x}{y}\right]
  \]
  
  Simplifying \( \frac{d}{dx}\left[\frac{x}{y}\right] \):
  
  \[
  \frac{d}{dx}\left[\frac{x}{y}\right] = \frac{y - x y'}{y^2}
  \]
  
  Therefore:
  
  \[
  \frac{d}{dx}\left[\log\left(\frac{x}{y}\right)\right] = \frac{y - x y'}{x}
  \]
  
  So, the derivative of the first term is:
  
  \[
  \ln(10) \cdot \frac{y - x y'}{x}
  \]

- Second term: \( \cos(xy) \)

  \[
  \frac{d}{dx}\left[\cos(xy)\right]
  \]
  
  Apply the chain rule:
  
  \[
  -\sin(xy) \cdot \frac{d}{dx}(xy) = -\sin(xy) \cdot (y + xy')
  \]

The derivative of the left side:

\[
\ln(10) \cdot \frac{y - x y'}{x} - \sin(xy) \cdot (y + xy')
\]

The derivative of the right side \( 5 \) is 0.

### Step 2: Combine and solve for \( y' \):

Set the derivatives equal:

\[
\ln(10) \cdot \frac{y - x y'}{x} - \sin(xy) \cdot (y + xy') = 0
\]

Factor out \( y' \):

\[
y'\left[ -\ln(10) + x\sin(xy)\right] = y\left[\sin(xy) - \frac{\ln(10)}{x}\right]
\]

Finally, solve for \( y' \):

\[
y' = \frac{y\left[\sin(xy) - \frac{\ln(10)}{x}\right]}{-\ln(10) + x\sin(xy)}
\]

This is the derivative \( y' \) for the given equation.

Do you want more details or have any questions? Here are some related questions to explore:

1. How would the derivative change if the argument of the logarithm was different?
2. What would happen if we used a different logarithmic base?
3. How does implicit differentiation work in cases where both \( x \) and \( y \) are in trigonometric functions?
4. How can we simplify expressions involving logarithmic and trigonometric functions before differentiation?
5. How would you interpret the result \( y' \) geometrically?

**Tip**: Always make sure to apply the chain rule correctly when differentiating composite functions involving products of \( x \) and \( y(x) \).

Learn how to find the derivative y' from the equation ln(10) log(x/y) + cos(xy) = 5 using implicit differentiation and the chain rule. Explore step-by-step solutions and understand the application of logarithmic and trigonometric functions in calculus. Suitable for advanced college students and learners of calculus.

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