Problem 21 - Calculus

Question

Questions \(20-29\) are about the derivative \(d y / d x=c b^{x}\). If the slope of \(\log x\) is \(1 / c x,\) find the slopes of \(\log (2 x)\) and \(\log \left(x^{2}\right)\) and \(\log \left(2^{2}\right)\)

Step-by-Step Solution

Answer

Slopes: \( \log(2x) = \frac{1}{cx} \), \( \log(x^2) = \frac{2}{cx} \), \( \log(4) = 0 \).

Step 1: Find the derivative of \( \log(2x) \)

The derivative of \( \log(2x) \) with respect to \( x \) is given by the chain rule. \( \log(2x) = \log(2) + \log(x) \).Therefore, the derivative is the same as the derivative of \( \log(x) \), which is \( \frac{1}{x} \).

Step 2: Apply the given slope condition

Given that the slope of \( \log(x) \) is \( \frac{1}{cx} \), we can apply this to the computed derivative above for both \( \log(2x) \) and \( \log(x) \). Therefore, the slope of \( \log(2x) \) is \( \frac{1}{cx} \).

Step 3: Find the derivative of \( \log(x^2) \)

For \( \log(x^2) \), use the logarithmic identity \( \log(x^2) = 2\log(x) \). The derivative is thus \( 2 \cdot \frac{1}{x} = \frac{2}{x} \).

Step 4: Apply the slope condition to \( \log(x^2) \)

Using the given slope condition, the slope of \( \log(x^2) \) becomes \( \frac{2}{cx} \).

Step 5: Find the derivative of \( \log(2^2) \)

Since \( \log(2^2) = \log(4) \), and \( \log(4) \) is a constant, its derivative is 0.

Step 6: Confirming results

Confirm that each step was completed accurately with the correct application of derivative rules and given conditions. Each derived slope fits within the logical constraints provided for \( \log(x) \).

Key Concepts

Chain rule

The chain rule is a fundamental tool in calculus essential for computing the derivatives of composite functions. Think of it as a way to "unwrap" nested functions by breaking them into manageable parts. If you have a function expressed as another function inside it, like \( f(g(x)) \), the chain rule allows you to differentiate it by focusing on the outer function first and then the inner one. Here's how it breaks down:

• Differentiate the outer function with respect to the inner function.
• Multiply that derivative by the derivative of the inner function.

Applying this rule to our problem, when finding the derivative of \( \log(2x) \), we recognized that \( \log(2x) \) contained the function \( 2x \). Thus, we differentiated \( \log(u) \) with respect to \( u \) and multiplied by the derivative of \( 2x \) with respect to \( x \). This process assures that every component of a compound expression is accounted for when differentiating.

Derivative of logarithmic functions

Logarithmic functions have their unique characteristics when it comes to differentiation. The function \( \log(x) \) has a well-known derivative of \( \frac{1}{x} \). This result stems from the natural properties of logarithms and how they stretch or compress transformations of a function. When dealing with modifications of \( \log(x) \) like \( \log(2x) \) or \( \log(x^2) \), a straightforward application of derivative rules can simplify the process.

• The derivative of \( \log(2x) \): Due to the property \( \log(2x) = \log(2) + \log(x) \), the \( \log(2) \) part vanishes when differentiating.
• The derivative of \( \log(x^2) \): This is simplified using the identity \( \log(x^2) = 2\log(x) \), making it easy to apply a constant multiple rule. Thus, the derivative is \( \frac{2}{x} \).
• The derivative of a constant logarithm like \( \log(4) \): Since it is constant, its derivative remains 0.

This approach ensures anyone can handle logarithmic functions' derivatives efficiently with precision.

Properties of logarithms

Understanding the properties of logarithms is crucial when manipulating and simplifying expressions. Logarithms have several properties that are particularly useful during differentiation:

• Logarithm of a product: \( \log(ab) = \log(a) + \log(b) \)
• Logarithm of a power: \( \log(a^b) = b\log(a) \)
• Constant logarithm: \( \log(a) \) for constant \( a \) is simply a number, whose derivative is zero.

These properties help simplify complicated logarithmic functions before applying differentiation rules. In our problem, reducing \( \log(2x) \) to \( \log(2) + \log(x) \) and \( \log(x^2) \) to \( 2\log(x) \) highlights how these fundamental properties streamline calculations. Remembering these rules can make seemingly complex calculus operations far more manageable.