What is the derivative of ln(x/y)?
Understand the Problem
The question is asking for the derivative of the natural logarithm of the ratio x/y. This involves applying the properties of logarithms and the rules of differentiation, such as the quotient rule.
Answer
$$ \frac{1}{x} - \frac{1}{y} \cdot \frac{dy}{dx} $$
Answer for screen readers
The derivative of $\ln\left(\frac{x}{y}\right)$ is given by:
$$ \frac{1}{x} - \frac{1}{y} \cdot \frac{dy}{dx} $$
Steps to Solve
- Apply the Logarithm Property
To find the derivative of the natural log of a ratio, use the property of logarithms:
$$ \ln\left(\frac{x}{y}\right) = \ln(x) - \ln(y) $$
This allows us to break the logarithm into two parts.
- Differentiate Each Component
Now, differentiate both sides with respect to $x$:
$$ \frac{d}{dx}\left(\ln\left(\frac{x}{y}\right)\right) = \frac{d}{dx} (\ln(x) - \ln(y)) $$
Using the derivatives of the logarithm, we have:
$$ \frac{d}{dx}(\ln(x)) = \frac{1}{x} $$
For $\ln(y)$, we need to use the chain rule since $y$ may depend on $x$:
$$ \frac{d}{dx}(\ln(y)) = \frac{1}{y} \cdot \frac{dy}{dx} $$
- Combine the Derivatives
Putting it all together, we have:
$$ \frac{d}{dx}\left(\ln\left(\frac{x}{y}\right)\right) = \frac{1}{x} - \frac{1}{y} \cdot \frac{dy}{dx} $$
This gives us the derivative of the natural logarithm of the ratio.
- Final Expression
Thus, the derivative of $\ln\left(\frac{x}{y}\right)$ is:
$$ \frac{1}{x} - \frac{1}{y} \cdot \frac{dy}{dx} $$
The derivative of $\ln\left(\frac{x}{y}\right)$ is given by:
$$ \frac{1}{x} - \frac{1}{y} \cdot \frac{dy}{dx} $$
More Information
The derivative of the natural logarithm of a ratio is useful in various applications in calculus and can help simplify many problems, especially those involving exponential functions and growth rates.
Tips
- Forgetting to apply the chain rule for $\ln(y)$ if $y$ is a function of $x$.
- Misapplying the logarithm property and not recognizing that $\ln\left(\frac{x}{y}\right)$ can be expressed as $\ln(x) - \ln(y)$.