Problem 54 - Calculus for Biology and Medicine

Question

Differentiate the functions with respect to the independent variable. $$ f(x)=\log \left(2 x^{2}-1\right) $$

Step-by-Step Solution

Answer

The derivative is \( \frac{4x}{2x^2 - 1} \).

Step 1: Identify the Function and the Rule Needed

The given function is \( f(x) = \log(2x^2 - 1) \). To differentiate this function, since it is a composition of functions (a logarithm of a polynomial), we will apply the chain rule.

Step 2: Differentiate the Outer Function

The outer function is \( \log(u) \) where \( u = 2x^2 - 1 \). The derivative of \( \log(u) \) with respect to \( u \) is \( \frac{1}{u} \).

Step 3: Differentiate the Inner Function

The inner function is \( u = 2x^2 - 1 \). The derivative of this with respect to \( x \) is \( \frac{d}{dx}(2x^2 - 1) = 4x \).

Step 4: Apply the Chain Rule

According to the chain rule, the derivative of the composite function \( \log(2x^2 - 1) \) is\[\frac{d}{dx}f(x) = \frac{1}{2x^2 - 1} \times 4x = \frac{4x}{2x^2 - 1}\]

Step 5: Simplify the Expression

The derivative of \( f(x) = \log(2x^2 - 1) \) simplifies to \( \frac{4x}{2x^2 - 1} \). This is the final simplified form of the derivative.

Key Concepts

Chain Rule

In calculus, the chain rule is a fundamental tool used to differentiate composite functions. It links the derivative of two or more interdependent functions, allowing us to find the derivative of the entire composition. The chain rule formula can be written as:
\[ \frac{d}{dx} [f(g(x))] = f'(g(x)) \cdot g'(x) \]
Here, the derivative of the outer function, evaluated at the inner function, is multiplied by the derivative of the inner function.

• Identify the outer and inner functions correctly.
• Differentiate the outer function with respect to the inner function.
• Differentiate the inner function with respect to the independent variable.
• Finally, multiply these derivatives accordingly.

Using the chain rule effectively can simplify the process of differentiation, especially when dealing with complex compositions.

Composite Functions

When we talk about composite functions, we refer to a situation where one function is inside another. This means the output of one function becomes the input for the second. Let's say you have two functions, \( g(x) \) and \( f(x) \). Then, the composite function is represented as \( f(g(x)) \).
For example, in the function \( \log(2x^2 - 1) \), \( 2x^2 - 1 \) is the inner function, while \( \log(u) \) is the outer function.

• Composite functions often require careful differentiation, as you deal with two layers.
• Use the chain rule to efficiently find derivatives of composite functions.
• It's crucial to change variables when needed, ensuring each function's derivative is accurately computed.

Mastering composite functions helps tackle problems involving complex variable transformations and operations.

Logarithmic Differentiation

Logarithmic differentiation is a technique that involves using properties of logarithms to differentiate functions that might be cumbersome to differentiate otherwise. It is especially useful for products, quotients, and powers of functions.
The method can be outlined as follows:

• Take the natural logarithm of both sides of the equation \( y = f(x) \).
• Use logarithmic identities to simplify and manipulate the function.
• Differentiate implicitly with respect to \( x \).
• Solve for \( \frac{dy}{dx} \) by exponentiating the result if necessary.

In the case of \( f(x) = \log(2x^2 - 1) \), logarithmic differentiation simplifies the expression by leveraging the natural logarithm's property of converting multiplication into addition. This often makes differentiation more straightforward and manageable.