Problem 28 - Calculus Single Variable

Question

Calculate the derivative with respect to \(x\) of the given expression. \(\log _{5}(5+2 x)\)

Step-by-Step Solution

Answer

The derivative is \( \frac{2}{(5+2x) \ln(5)} \).

Step 1: Identify the structure of the function

The expression given is a logarithmic function with base 5: \( \log_{5}(5+2x) \). The differentiation will involve using the change of base formula and chain rule.

Step 2: Change the base of the logarithm

To differentiate a logarithm with any base, use the change of base formula. Convert \( \log_{5}(5+2x) \) into natural logarithms (base \(e\)): \[ \log_{5}(5+2x) = \frac{\ln(5+2x)}{\ln(5)} \].

Step 3: Differentiate using the chain rule

The expression is now \( \frac{\ln(5+2x)}{\ln(5)} \). We can factor out \( \frac{1}{\ln(5)} \) since it is constant. Then, differentiate \( \ln(5+2x) \): \[ \frac{d}{dx}\ln(5+2x) = \frac{1}{5+2x} \cdot \frac{d}{dx}(5+2x) = \frac{1}{5+2x} \cdot 2 \].

Step 4: Combine to find the final derivative

Multiply the differentiated result by the constant we factored out: \[ \frac{d}{dx}\log_{5}(5+2x) = \frac{1}{\ln(5)} \cdot \frac{2}{5+2x} \].

Step 5: Simplify the expression

The final expression for the derivative is \[ \frac{2}{(5+2x) \ln(5)} \].

Key Concepts

Logarithmic Differentiation

Logarithmic differentiation is a technique used to find the derivative of functions that are logarithmic in nature. This technique is especially useful when dealing with logs of varying bases. The process involves taking the natural logarithm of both sides of an equation before differentiating. This can simplify the differentiation process, especially when exponents or products are involved. In the context of our exercise, we had a logarithmic function with base 5, and we needed to differentiate it with respect to its argument.
By employing logarithmic differentiation, we converted it into a form that was more manageable by using the natural logarithm, making it easier to apply differentiation rules. This approach demonstrates the power of logarithmic functions when paired with differentiation, allowing us to break complex expressions down into simpler parts.

Change of Base Formula

The change of base formula is an essential mathematical tool for working with logarithms. When you need to transform a logarithm of one base into another, the formula is:

• \( \log_b(a) = \frac{\ln(a)}{\ln(b)} \)

This formula allows us to change the base of a logarithm to any desired base, commonly into the natural logarithm (base \(e\)), which is more frequently used in calculus due to its natural properties.
In our example with \( \log_{5}(5+2x) \), we used this formula to shift from base 5 to base \(e\). It turned the problem into a simpler form that leverages the properties of natural logarithms, making the upcoming differentiation steps more straightforward.

Chain Rule

The chain rule is a fundamental derivative rule in calculus used to differentiate composite functions. When a function is the combination of two functions, the chain rule facilitates differentiation, even if layers of functions are involved.
In simpler terms, when differentiating a composite function \( f(g(x)) \), the chain rule provides the derivative as:

• \( \frac{d}{dx}f(g(x)) = f'(g(x)) \cdot g'(x) \)

In our exercise, after applying the change of base formula, we needed to differentiate \( \ln(5+2x) \). The chain rule helps by identifying \( g(x) = 5 + 2x \), and thus, differentiating \( \ln(g(x)) \) necessitates finding \( \frac{1}{g(x)} \cdot g'(x) \). This step ensures that each nested function is appropriately differentiated based on its relationship to the inner function.

Natural Logarithms

Natural logarithms are logarithms with the base \(e\), an irrational number approximately equal to 2.71828. They are particularly important in calculus because of their unique properties that simplify many calculus operations, especially differentiation.
When differentiating expressions involving natural logs, the rule is straightforward:

• \( \frac{d}{dx}(\ln(x)) = \frac{1}{x} \)

This simplicity is why natural logarithms are preferred when applying the change of base formula or during processes like logarithmic differentiation.
In our example, transforming the base 5 logarithm to a natural logarithm allows us to use these simple differentiation rules. By converting to \( \ln(5+2x) \), we capitalized on these properties, ultimately making the derivative calculation a more direct process. This showcases the efficiency and effectiveness of natural logs in calculus, streamlining operations and making complex differentiations approachable.