Question

In Exercises, find the derivative of the function. $$ y=\ln \frac{x}{x+1} $$

Short answer

The derivative of the function \(y=\ln \frac{x} {x+1}\) is \(y' = \frac {1} {x} - \frac{1} {x+1}\)

Step-by-step solution

Expand the function

Using the properties of the logarithms, the given function can be simplified by expressing it as a difference of two logs: \(y=\ln x - \ln (x+1)\).

Apply the derivative rules for each term

Differentiate each term separately using derivative log rule and chain rule. The derivative of \(\ln x\) is \(\frac{1} {x}\), and for \(\ln (x+1)\) you'll need to use the chain rule, which states that the derivative of \(\ln u\) is \(\frac{u'} {u}\), where \(u' \) is the derivative of \(u\). Here, \(u = x + 1\), so the derivative, \(u'\), will be 1. Thus \(\frac {d (\ln (x+1))} {dx} = \frac {1} {x+1}\).

Subtract the derivative of the second term from the first

Now the derivative of the function will be the derivative of the first term minused the derivative of the second term: \(y' = \frac {1} {x} - \frac{1} {x+1}\)

Key concepts

Derivative

In calculus, a derivative represents how a function changes as its input changes. It tells us the rate of change or the slope of the tangent line at any point on a graph.
Derivatives are central to differential calculus and are used in almost every aspect of applied mathematics.
To find a derivative, you apply specific rules, depending on the function type, to get that rate of change.For any basic function like \(f(x) = x^2\), the derivative \(f'(x)\) tells us how fast \(f(x)\) changes as \(x\) changes.
For instance, the derivative of \(f(x) = x^2\) is \(2x\), indicating that at any point \(x\), the slope or rate of change is \(2x\).By understanding derivatives, we can predict behavior, find optimizations, and understand the nature of the functions we're working with, especially in more complex scenarios.

Logarithmic Differentiation

Logarithmic differentiation is a clever method to differentiate functions that are not easily manipulated by standard rules. It's especially useful for complex products, quotients, or powers.### How it Works- First, take the natural logarithm (\(\ln\)) of both sides of the equation you're working with.- Use the properties of logarithms to simplify the expression. This often turns products into sums and quotients into differences.
For example, if you have the function \(y = \ln \frac{x}{x+1}\), it can be rewritten using logarithmic properties as \(\ln y = \ln x - \ln(x+1)\).
This makes it easier to differentiate each part separately.### Benefits- Simplifies the differentiation process.- Provides a clearer view for derivation in complex or compound functions.- Avoids cumbersome algebraic manipulation once the function is in a logarithmic form.By mastering logarithmic differentiation, you can tackle challenging calculus problems with confidence.

Chain Rule

The chain rule is a fundamental principle in calculus used to differentiate composite functions. It allows us to find the derivative of a function that is composed of two or more functions.### Applying the Chain Rule- Identify the outer function and inner function. In a composite function like \(f(g(x))\), \(f\) is the outer function and \(g\) is the inner function.- Differentiate the outer function, but keep the inner function unchanged initially.- Then, multiply by the derivative of the inner function.For example, if we have a function like \(h(x) = (x+1)^2\), the chain rule tells us to first differentiate \(u^2\) (where \(u = x+1\)), which gives \(2u\).
Then, multiply it by the derivative of the inner function \(u = x + 1\), which is 1, resulting in a final derivative of \(2(x+1) \times 1 = 2(x+1)\).The chain rule is indispensable when handling functions within functions, providing a systematic way to find derivatives of any composed function.