Question

For the following exercises, find the derivative \(\frac{d y}{d x}\). $$ y=\ln (2 x+1) $$

Short answer

The derivative of \( y = \ln(2x+1) \) is \( \frac{2}{2x+1} \).

Step-by-step solution

Identify the Rule

To find the derivative of a natural logarithmic function, we use the chain rule. The derivative of \( \ln(u) \) is \( \frac{1}{u} \times u' \), where \( u \) is a function of \( x \).

Define the Inner Function

In our problem, \( u = 2x + 1 \). So, we must differentiate \( u \) with respect to \( x \).

Differentiate the Inner Function

The derivative of the inner function \( u = 2x + 1 \) with respect to \( x \) is \( u' = \frac{d}{dx}(2x + 1) = 2 \).

Apply the Chain Rule

Using the chain rule for differentiation, substitute \( u \) and \( u' \) into the general rule: \( \frac{dy}{dx} = \frac{1}{u} \cdot u' = \frac{1}{2x+1} \cdot 2 = \frac{2}{2x+1} \).

Simplify the Derivative

The expression \( \frac{2}{2x+1} \) is already in its simplest form, representing the derivative of \( y = \ln(2x + 1) \).

Key concepts

Understanding Derivatives

In calculus, a derivative represents the rate at which a function is changing at any given point. It gives us the slope of the tangent line to the curve at a particular point. Derivatives are fundamental because they help us analyze the behavior of functions.
For any function \( y = f(x) \), the derivative, denoted as \( \frac{dy}{dx} \), tells us how \( y \) changes with a small change in \( x \). This rate of change can be understood as the 'instantaneous speed' of the function as \( x \) changes.

• In simple terms, if the derivative is positive, the function is increasing.
• If the derivative is negative, the function is decreasing.
• If the derivative is zero, the function may have a local maximum or minimum at that point.

Understanding derivatives allows us to solve problems involving rates, such as velocity and acceleration. It is also crucial for finding maxima and minima in optimization problems.

The Chain Rule Explained

The chain rule is a formula for computing the derivative of a composition of functions. When dealing with nested functions—functions inside other functions—the chain rule is the tool that allows us to differentiate them.
If you have two functions, \( f \) and \( g \), where \( y = f(g(x)) \), the chain rule states that the derivative of \( y \) with respect to \( x \) is \( f'(g(x)) \cdot g'(x) \). This means you differentiate the outer function, keeping the inner function unchanged, and then multiply by the derivative of the inner function.

• Identify the outer function and differentiate it.
• Differential the inner function.
• Multiply both derivatives.

This is especially useful when the function is too complex to be tackled with simple differentiation. For example, with the function \( y = \ln(2x + 1) \), using the chain rule simplifies differentiation to \( \frac{2}{2x+1} \). This systematic method ensures we correctly handle the differentiation of compositions.

Natural Logarithmic Functions

Natural logarithmic functions are a special type of logarithm that use the constant \( e \) as the base. The natural logarithm \( \ln(x) \) is the inverse function of the exponential function \( e^x \). It is widely used in calculus and is essential for solving problems involving growth and decay.The derivative of the natural logarithm, \( y = \ln(u) \), is given by \( \frac{1}{u} \cdot u' \). This formula is derived from the properties of logarithms. When dealing with natural logarithmic functions,
you often apply the chain rule to find the derivative correctly:

• Identify the expression inside the logarithm.
• Differential that inner expression.
• Apply the derivative formula for natural logs.

Understanding natural logs and their derivatives is vital for working through problems involving exponential growth, interest calculations, and more in advanced calculus settings.