Question

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

Short answer

The derivative \( \frac{dy}{dx} = \frac{7}{x} \).

Step-by-step solution

Rewrite the Function

First, observe the given function \( y = \ln((4x)^7) \). Using the logarithmic property \( \ln(a^b) = b \ln(a) \), rewrite the function as: \[ y = 7 \ln(4x) \].

Identify the Inner Function

The function now is \( y = 7\ln(4x) \). Identify the inner function, which is the argument of the logarithm. Here, \( u = 4x \), and consequently \( \ln(u) \).

Differentiate the Inner Function

Differentiate the inner function \( u = 4x \). The derivative \( \frac{du}{dx} = 4 \), since the derivative of \( 4x \) with respect to \( x \) is 4.

Differentiate the Outer Function

Using the chain rule, differentiate the outer function. For \( y = 7\ln(u) \), the derivative with respect to \( u \) is \( \frac{d}{du}[7\ln(u)] = \frac{7}{u} \).

Apply the Chain Rule

Apply the chain rule to find the derivative \( \frac{dy}{dx} \). Multiply the derivative of the outer function by the derivative of the inner function: \[ \frac{dy}{dx} = \frac{7}{4x} \times 4 = \frac{28}{4x} \].

Simplify the Derivative

The expression \( \frac{28}{4x} \) simplifies to \( \frac{7}{x} \). Thus, the derivative of the function \( y \) with respect to \( x \) is \( \frac{dy}{dx} = \frac{7}{x} \).

Key concepts

Chain Rule

The chain rule is one of the most essential techniques in calculus, used particularly when dealing with composite functions—functions defined as a combination of two or more other functions. The chain rule enables us to differentiate these composite functions systematically. Let’s break it down further into relatable steps.

• First, identify the outer function. In our example, the outer function is the natural logarithm function, expressed as \( y = \ln(u) \).
• Next, identify the inner function. For the original function \( (4x)^7 \), we simplified it to \( u = 4x \) after using logarithmic properties.

Once both functions are identified, differentiate the inner function and the outer function separately:

• The derivative of the inner function \( u = 4x \) is \( \frac{du}{dx} = 4 \).
• The derivative of the outer function \( 7\ln(u) \) with respect to \( u \) is \( \frac{7}{u} \).

Now, apply the chain rule: multiply these derivatives together to get the derivative with respect to \( x \):

• \( \frac{dy}{dx} = \frac{7}{4x} \times 4 = \frac{28}{4x} \).

Finally, simplify this expression to find the overall derivative.

Logarithmic Differentiation

Logarithmic differentiation is a technique particularly useful when differentiating functions that involve powers and products, like \( (4x)^7 \). Instead of directly differentiating, we first use logarithmic identities to simplify the expression, which can make the differentiation steps more manageable.
Here, by recognizing the power inside the logarithm, \( \ln((4x)^7) \) becomes \( 7\ln(4x) \). This step utilizes the property \( \ln(a^b) = b\ln(a) \). By rewriting the function, we effectively transformed a power function into a function manageable by basic differentiation rules.
The primary advantage of logarithmic differentiation here is:

• Breaking down complex roles (like the repeated multiplication across powers) into linear forms.
• Simplifying products or quotients by converting them to sums and differences using logarithms.

Once simplified, we can focus on the derivative rules of simpler functions, using our standard differentiation techniques as seen in the problem solution.

Function Simplification

Function simplification is a crucial preprocessing step when dealing with complex derivatives. It refers to the process of transforming a complicated function into a simpler form, making the function easier to differentiate. In our exercise, we started with \( y = \ln((4x)^7) \). By applying the logarithmic property \( \ln(a^b) = b\ln(a) \), the function becomes easier to handle: \( y = 7\ln(4x) \).
This simplification allowed for clearer identification of the inner and outer functions, essential to apply the chain rule effectively. Simplifying functions not only reduces errors in subsequent differentiation but also:

• Clarifies the structure of a function for easier visualization of layers (inner vs. outer functions in this case).
• Turns multiplication and power operations into additions and simple multiplications, vastly simplifying calculations.

In the end, thorough simplification can save significant time and effort, especially useful during exams or solving multiple derivative problems.