Problem 306 - Calculus Volume 1

Question

For the following exercises, find the derivative dy/dx. (You can use a calculator to plot the function and the derivative to confirm that it is correct.) [T] \(y=\ln \left((4 x)^{7}\right)\)

Step-by-Step Solution

Answer

The derivative is \( \frac{7}{x} \).

Step 1: Identify the function

We are given the function \( y = \ln((4x)^7) \). Here, the function inside the logarithm is \((4x)^7\).

Step 2: Apply the logarithm power rule

Utilize the property of logarithms that states \( \ln(a^b) = b \cdot \ln(a) \). Applying this to our function, we get \( y = 7 \cdot \ln(4x) \).

Step 3: Differentiate using the chain rule

Differentiate \( y = 7 \cdot \ln(4x) \). The derivative of \( \ln(u) \) with respect to \( x \) is \( \frac{1}{u} \cdot \frac{du}{dx} \). Here, \( u = 4x \), so \( \frac{du}{dx} = 4 \). Thus, \( \frac{dy}{dx} = 7 \cdot \frac{1}{4x} \cdot 4 = \frac{7}{x} \).

Step 4: Write the final expression

The derivative \( \frac{dy}{dx} \) of \( y = \ln((4x)^7) \) is \( \frac{7}{x} \).

Key Concepts

Logarithm

A logarithm is a mathematical operation that helps us understand repeated multiplication, yet in reverse. For example, when the base 10 logarithm is used, it tells us how many times we multiply 10 by itself to get another number. Specifically, \( \ln(a^b) = b \cdot \ln(a) \) is a useful rule when working with logarithms. It states that the logarithm of a power can be expressed as the exponent times the logarithm of the base. This property simplifies complex exponential relationships into easier-to-handle expressions.
Applying it to our exercise: \( y = \ln((4x)^7) \) becomes \( y = 7 \cdot \ln(4x) \). This simplification is crucial before differentiation, as it makes the following steps much simpler.

Chain Rule

The chain rule is a fundamental technique in calculus for differentiating compositions of functions. If you have a composite function like \( y = f(g(x)) \), the chain rule allows you to differentiate by finding the derivative of the outer function and multiplying it by the derivative of the inner function.
In this exercise, our function is \( y = 7 \cdot \ln(4x) \). By setting \( u = 4x \), we identify the inner function. The corresponding outer function becomes based on \( \ln(u) \). When differentiating \( \ln(u) \), we apply the chain rule: \( \frac{d}{dx}(\ln(u)) = \frac{1}{u} \cdot \frac{du}{dx} \). Here, \( u = 4x \) which gives us \( \frac{du}{dx} = 4 \). Multiplying these, we apply the chain rule effectively.

Differentiation

Differentiation focuses on finding rates of change or slopes of functions. In this particular exercise, we are seeking the derivative of \( y = \ln((4x)^7) \).
We've already simplified it to \( y = 7 \cdot \ln(4x) \). The differentiation process uses the property that \( \frac{d}{dx}(\ln(u)) = \frac{1}{u} \cdot \frac{du}{dx} \). Once the inner function \( u \) is defined (where in our case, \( u = 4x \)), and \( \frac{du}{dx} = 4 \), we can compute the derivative: \( \frac{dy}{dx} = 7 \cdot \frac{1}{4x} \cdot 4 = \frac{7}{x} \).
This process highlights how simplification can make differentiation more straightforward and accurate.

Calculus Exercises

Calculus exercises serve multiple educational purposes, primarily strengthening our understanding of function behavior, rates of change, and limits. Exercises like \( y = \ln((4x)^7) \) assess your ability to manipulate, simplify, and differentiate functions accurately.

• First, look for simplification opportunities, like applying logarithmic properties.
• Apply pertinent rules such as the chain rule for nested functions.
• Check results using calculators or graphs to confirm they make sense.

By practicing these steps, students build the skills needed to tackle increasingly complex problems, paving the way for deeper calculus comprehension.