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.) $$ \text { [T] } y=7 \ln (4 x) $$

Short answer

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

Step-by-step solution

Identify the Derivative Formula

The function given is \( y = 7 \ln{(4x)} \). To find the derivative, we will use the chain rule, which is a method to differentiate composite functions. For a function \( f(g(x)) \), the derivative is \( f'(g(x))g'(x) \).

Differentiate the Outer Function

First, differentiate the outer function \( 7 \ln{(u)} \) with respect to \( u \). The derivative of \( \ln{(u)} \) is \( \frac{1}{u} \). Thus, the derivative of \( 7 \ln{(u)} \) is \( \frac{7}{u} \).

Differentiate the Inner Function

The inner function is \( u = 4x \). Differentiate \( 4x \) with respect to \( x \), which is simply \( 4 \).

Apply the Chain Rule

Apply the chain rule from Step 1. Multiply the derivative of the outer function by the derivative of the inner function: \( \frac{7}{4x} \times 4 \).

Simplify the Expression

Simplify \( \frac{7}{4x} \times 4 = \frac{28}{4x} = \frac{7}{x} \). This is the derivative \( \frac{dy}{dx} \).

Key concepts

Chain Rule

The chain rule is a fundamental technique in calculus used to find the derivative of composite functions. A composite function is one where a function is applied inside another function, like nesting doll layers. In simple terms, if you use two functions, say \( f \) and \( g \), to create a new function \( f(g(x)) \), the derivative requires the chain rule. Here's a simple breakdown of how it works:

• Differentiation involves finding how quickly a value changes. For composite functions, it might not be obvious at first glance.
• Apply the chain rule by first differentiating the outside function, ignoring the inside portion momentarily. Let's say the outer function is \( f(u) \) where \( u=g(x) \).
• Next, differentiate the inner function, \( g(x) \). Multiply these two derivatives together. This multiplication captures how changes in \( x \) affect \( u \), and subsequently \( f \).

This method ensures that all parts of the composite function are correctly considered, giving you the complete derivative. If you have the function from the exercise, \( y = 7 \ln(4x) \), the outer function is \( 7 \ln u \) and the inner is \( u = 4x \). This leads to the derivative: first \( 7/u \), then \( 4 \), which when multiplied yields \( \frac{7}{x} \).
Step by step approaches using the chain rule are invaluable for tackling more complex derivative challenges.

Composite Functions

Composite functions are the building blocks of more complex mathematical expressions. They appear when you combine two simpler functions into one, using the output of one as the input for the other.
In mathematical notation, a composite function is usually written as \( f(g(x)) \), which means you first apply the function \( g \) and then apply the function \( f \).

• Understanding composite functions starts with recognizing the order of operations: the function order matters because it determines how inputs are transformed.
• In the given exercise, inside \( y=7 \ln(4x) \), the "\( 4x \)" modifies the logarithmic function \( \ln \), creating the composite structure.
• Breaking these functions apart helps in utilizing calculus techniques like the chain rule effectively.

Through composite functions, complex interactions can be explored?providing a glimpse into deeper relationships within mathematics. This dissection helps simplify the finding of derivatives by identifying inner and outer functions.

Logarithmic Differentiation

Logarithmic differentiation is a strategic approach to finding derivatives of functions involving logarithms. It often simplifies the differentiation process, especially when dealing with products or quotients of functions.

• When you see a logarithm, remember their properties: they transform multiplicative relationships into additive ones. This simplification is key in differentiation.
• To differentiate a logarithm, use the basic derivative formula for \( \ln(x) \), which is \( \frac{1}{x} \). This forms the backbone for more complex applications.
• In \( y = 7 \ln(4x) \), using logarithmic differentiation, the \( \ln(4x) \) becomes \( \frac{1}{4x} \) when differentiated. Multiply by the constant 7, combined with any additional derivatives from the chain rule application, leading to the final form \( \frac{7}{x} \).

This process highlights the power of logarithmic differentiation: its ability to transform challenging derivative calculations into straightforward, manageable steps. Mastery in logarithmic differentiation not only helps in specific problems but broadens one's ability to tackle a wide range of calculus challenges efficiently. Understanding this method is essential for anyone looking to deepen their calculus toolkit.