Problem 7 - Calculus

Question

Find the derivative \(d y / d x\) in \(1-10\). $$ y=\ln (\sin x) $$

Step-by-Step Solution

Answer

The derivative is \( \frac{d y}{d x} = \cot x \).

Step 1: Identify the function and use chain rule

The function given is \( y = \ln(\sin x) \). Notice that \( y \) is a composition of functions: the natural logarithm function \( \ln u \) and \( u = \sin x \). We'll apply the chain rule, which states that if \( y = g(f(x)) \), then \( \frac{d y}{d x} = \frac{d g}{d f} \cdot \frac{d f}{d x} \).

Step 2: Differentiate the outer function

The outer function is \( \ln(u) \), where \( u = \sin x \). The derivative of \( \ln(u) \) with respect to \( u \) is \( \frac{1}{u} \). Thus, \( \frac{d}{d u}(\ln u) = \frac{1}{u} \).

Step 3: Differentiate the inner function

The inner function is \( \sin x \). The derivative of \( \sin x \) with respect to \( x \) is \( \cos x \). Thus, \( \frac{d}{d x}(\sin x) = \cos x \).

Step 4: Apply the chain rule

By the chain rule, we find \( \frac{d y}{d x} = \frac{d \ln(\sin x)}{d \sin x} \cdot \frac{d \sin x}{d x} \). Substituting the derivatives from the previous steps, we have \( \frac{d y}{d x} = \frac{1}{\sin x} \cdot \cos x \). Thus, \( \frac{d y}{d x} = \frac{\cos x}{\sin x} \).

Step 5: Simplify the expression

The expression \( \frac{\cos x}{\sin x} \) can be simplified to \( \cot x \), which is the cotangent of \( x \). Thus, the derivative \( \frac{d y}{d x} = \cot x \).

Key Concepts

Chain Rule

When dealing with composite functions, such as when one function is inside another, the **Chain Rule** becomes essential for differentiation. The chain rule is a fundamental technique in calculus for finding the derivative of a composition of functions. If you have a composite function like \( y = g(f(x)) \), the derivative is calculated as \( \frac{d y}{d x} = \frac{d g}{d f} \times \frac{d f}{d x} \). This means you first derive the outer function with respect to the inner function, and then derive the inner function itself with respect to \( x \).

• The outer function is \( g(u) \), where \( u = f(x) \).
• The inner function is \( f(x) \).

For example, in the problem \( y = \ln(\sin x) \), the outer function is \( \ln u \), and the inner function is \( u = \sin x \). Using the chain rule helps streamline the process of differentiation, especially when functions get complex. This makes it easier to tackle problems involving nested functions without getting bogged down in cumbersome calculations.

Natural Logarithm

The **Natural Logarithm**, commonly denoted as \( \ln \), is a logarithm with the base \( e \), where \( e \approx 2.71828 \). It is a special type of logarithm that is particularly useful in calculus due to its nice properties.
Some key features include:

• The derivative of \( \ln(u) \) with respect to \( u \) is given by \( \frac{1}{u} \).
• This derivative is crucial when applying the chain rule, as seen in the problem \( y = \ln(\sin x) \). Here, the derivative of the natural logarithm plays a critical role in simplifying the overall differentiation process.

Because of its significance in growth processes and natural phenomena, the natural logarithm is frequently encountered in various branches of mathematics, physics, and engineering. Understanding the natural logarithm and its properties allows one to better navigate problems involving exponential growth and decay.

Trigonometric Functions

**Trigonometric Functions**, such as \( \sin x \) and \( \cos x \), are vital tools in mathematics, especially in calculus and geometry. These functions relate the angles of a triangle to the ratios of its sides in a right triangle.
In calculus, the derivatives of these functions are fundamental:

• The derivative of \( \sin x \) is \( \cos x \).
• The derivative of \( \cos x \) is \( -\sin x \).

In the given problem, \( y = \ln(\sin x) \), we differentiate \( \sin x \) as part of the chain rule to find \( \frac{d}{dx}(\sin x) = \cos x \). This step is essential for applying the chain rule correctly. By understanding these derivatives, you can handle a wide range of problems in calculus involving periodic phenomena, oscillations, and more.

Function Composition

**Function Composition** means combining two or more functions such that the output of one function becomes the input of another. When calculating derivatives of composed functions, recognizing the composition structure is key.
For instance, if you have \( y = \ln(\sin x) \), you're dealing with:

• The outer function: \( \ln(u) \).
• The inner function: \( u = \sin x \).

Understanding function composition lets you clearly organize complex expressions and apply the chain rule effectively. This decomposition simplifies handling functions that are interdependent and maintains clarity throughout the derivation process. Function composition is everywhere in mathematics, making it a crucial concept to master as you encounter more intricate equations and real-world applications.