Problem 10 - Calculus

Question

Find the derivative \(d y / d x\) in \(1-10\). $$ y=\ln \left((4 x)^{7}\right) $$

Step-by-Step Solution

Answer

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

Step 1: Rewrite the Expression Using Properties of Logarithms

The expression \( y = \ln((4x)^7) \) can be simplified using the logarithm power rule. This rule states that \( \ln(a^b) = b \cdot \ln(a) \). Thus, we can rewrite the expression as \( y = 7 \cdot \ln(4x) \).

Step 2: Apply the Chain Rule for Differentiation

To find \( \frac{dy}{dx} \), we need to differentiate \( y = 7 \cdot \ln(4x) \). Since \( 4x \) is inside the logarithm, we'll apply the chain rule. We first differentiate the outer function, obtaining \( 7 \cdot \frac{d}{dx} \ln(4x) = 7 \cdot \frac{1}{4x} \), and then multiply this by the derivative of the inner function \(4x\).

Step 3: Differentiate the Inner Function

Differentiate \( 4x \) with respect to \( x \). The derivative of \(4x\) is 4.

Step 4: Multiply Derivatives According to the Chain Rule

Combine the derivatives from steps 2 and 3: \( 7 \cdot \frac{1}{4x} \cdot 4 = \frac{28}{4x} \).

Step 5: Simplify the Expression

Simplify \( \frac{28}{4x} \) to \( \frac{7}{x} \) by dividing both the numerator and the denominator by 4.

Key Concepts

Chain Rule

The chain rule is a fundamental concept used in calculus to differentiate composite functions, which are functions made up of two or more simpler functions. When we have a function inside another function, this rule helps us find the derivative.

The chain rule states: if a function is composed as \( f(g(x)) \), then its derivative is given by \((f \, \circ \, g)'(x) = f'(g(x)) \cdot g'(x)\). This means that we first

• differentiate the outer function \( f \) with respect to the inner function \( g(x) \),
• then multiply it by the derivative of the inner function \( g(x) \) with respect to \( x \).

In our exercise, the chain rule is applied when finding the derivative of \( \ln(4x) \). We treat \( \ln \) as the outer function and \( 4x \) as the inner function. By first differentiating the outer logarithm as \( \frac{1}{4x} \) and then multiplying by the derivative of the inner \( 4x \), which is 4, we effectively use the chain rule.

Logarithmic Differentiation

Logarithmic differentiation is a technique that simplifies the differentiation process of expressions where a variable appears both as a base and an exponent. This method involves using logarithms to decompose complex products and powers that make direct differentiation cumbersome.

For a function of the form \( y = a^b(x) \), you can apply the natural logarithm on both sides to

• transform the power into a product,
• utilize the properties of logarithms to simplify the differentiation.

In our case, differentiating \( y = \ln((4x)^7) \) becomes easier when rewritten using the property \( \ln(a^b) = b \cdot \ln(a) \). This enabled us to rewrite and differentiate the expression as \( 7 \cdot \ln(4x) \). By simplifying the power first, we apply the differentiation rules directly and more easily.

Power Rule

The power rule is a straightforward rule for finding the derivative of power functions of the form \( x^n \). It's a fundamental tool in calculus and significantly aids in simplifying the differentiation process.

The power rule formula states: if \( y = x^n \), then \( \frac{dy}{dx} = n \cdot x^{n-1} \). It tells us to

• bring down the exponent as a coefficient,
• then subtract one from the exponent.

In our derivation of \( \ln((4x)^7) \), rewriting the expression using logarithmic properties initially simplifies our task to working with a linear form, rather than a power expression directly. However, understanding how the power rule operates helps us appreciate why the logarithmic transformation facilitates easy differentiation here.

Calculus Problem Solving

Solving calculus problems involves understanding the appropriate techniques and applying them systematically. This includes identifying the function presentation, selecting suitable differentiation methods, and simplifying expressions.

The problem-solving steps often include:

• identifying the expression format, often involving a mix of rules like the chain rule, product rule, and power rule,
• rewriting complex expressions using identities like logarithmic and trigonometric identities,
• selectively applying differentiation techniques to obtain the derivative.

In our exercise, the process began with applying the properties of logarithms to rewrite the function, followed by systematically applying the chain rule. Lastly, simplification of the obtained derivative ensures clarity and correctness in the solution. Practice enhances proficiency in recognizing and applying these problem-solving strategies.