Problem 64 - Calculus and its applications

Question

Use the Chain Rule, implicit differentiation, and other techniques to differentiate each function given. $$ y=\log _{a} f(x), \text { for } f(x) \text { positive } $$

Step-by-Step Solution

Answer

The derivative is \(\frac{f'(x)}{f(x) \ln(a)}\).

Step 1: Recall the Formula for Logarithm Differentiation

We start by recalling the derivative of a logarithm with a general base \(a\):\[\frac{d}{dx} [\log_a(x)] = \frac{1}{x \ln(a)}\]This will guide us in differentiating \(y = \log_a[f(x)]\).

Step 2: Apply Chain Rule for Differentiation

The Chain Rule states that the derivative of a function \(y = f(g(x))\) is \(f'(g(x)) \cdot g'(x)\). For the given log function, we differentiate the outer function and multiply by the derivative of the inner function:\[\frac{d}{dx} \left[ y \right] = \frac{1}{f(x) \ln(a)} \cdot \frac{d}{dx} [f(x)]\]

Step 3: Differentiate the Inner Function

Now we differentiate the inner function \(f(x)\) with respect to \(x\):\[\frac{d}{dx} [f(x)] = f'(x)\]

Step 4: Combine Results

Substituting the derivative of \(f(x)\) from Step 3 back into the expression from Step 2, the derivative of \(y\) is:\[\frac{d}{dx} [y] = \frac{1}{f(x) \ln(a)} \cdot f'(x)\]This is the derivative of \(y = \log_a[f(x)]\).

Key Concepts

Logarithmic Differentiation

Logarithmic differentiation is a versatile technique often used to simplify the differentiation of complex functions. This technique is particularly helpful when dealing with products, quotients, or powers. The core idea is to take the logarithm of the function and then differentiate. It allows us to transform multiplicative relationships into additive ones, which are generally easier to manage.

• To use logarithmic differentiation, first take the natural logarithm of both sides of the equation.
• Differentiate using the properties of logarithms, such as the power rule: \[\ln(x^n) = n \cdot \ln(x)\].
• Simplify the resulting expression, if possible.

Logarithmic differentiation is advantageous because it breaks down intricate expressions into simpler components, making the subsequent steps of finding derivatives easier.

Implicit Differentiation

Implicit differentiation is a powerful method used when a function is not solely expressed as "y =" but rather involves y intertwining with x in a more complex equation. This technique is particularly useful for finding derivatives when variables are mixed and not easily separable.
When employing implicit differentiation, follow these steps:

• Differentiate both sides of the equation with respect to x. Remember to treat y as a function of x and apply the chain rule accordingly.
• Every time you differentiate a term with y in it, remember to multiply by \( \frac{dy}{dx} \), since y is implicitly a function of x.
• Solve for \( \frac{dy}{dx} \) after differentiating the entire equation.

This technique allows us to find \( \frac{dy}{dx} \) without explicitly solving for y in terms of x, making it suitable for complex equations that define y implicitly.

Derivative Calculation Techniques

Derivative calculation techniques encompass a variety of rules and methods used to find derivatives of functions. Some of the foundational techniques include:

• Power Rule: For any function of the form \(x^n\), the derivative is \(n \cdot x^{n-1}\).
• Product Rule: When differentiating a product of two functions, apply: \[ (uv)' = u'v + uv' \] where \(u\) and \(v\) are functions of x.
• Quotient Rule: For a quotient \(\frac{u}{v}\), use:\[\left( \frac{u}{v} \right)' = \frac{u'v - uv'}{v^2}\]
• Chain Rule: Used for composite functions \(f(g(x))\), differentiate using:\[ f'(g(x)) \cdot g'(x) \]

Understanding these rules is crucial for tackling calculus problems efficiently. Often, these techniques need to be used in combination to handle more complex derivatives as seen in the exercise involving the chain rule and logarithmic functions.