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

Short answer

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

Step-by-step solution

Understand the Function

The function given is \( y = \ln(\sin x) \). We need to find the derivative \( \frac{dy}{dx} \). This requires using the chain rule and the derivative properties of the natural logarithm and trigonometric functions.

Derivative of the Natural Logarithm Function

Recall the derivative of \( \ln(u) \) is \( \frac{1}{u} \frac{du}{dx} \). Here, \( u = \sin x \). So, the derivative \( \frac{d}{dx}[\ln(\sin x)] = \frac{1}{\sin x} \frac{d}{dx}[\sin x] \).

Find the Derivative of \(\sin x\)

The derivative of \( \sin x \) is \( \cos x \). Therefore, \( \frac{d}{dx}[\sin x] = \cos x \).

Apply the Chain Rule

Using the chain rule from Step 2, substitute \( \frac{d}{dx}[\sin x] = \cos x \) into the derivative formula: \( \frac{1}{\sin x} \cdot \cos x = \frac{\cos x}{\sin x} \).

Simplify the Expression

The expression \( \frac{\cos x}{\sin x} \) simplifies to \( \cot x \) because \( \cot x = \frac{\cos x}{\sin x} \).

Final Answer Verification

Therefore, the derivative \( \frac{dy}{dx} \) of \( y = \ln(\sin x) \) is \( \cot x \). You can plot \( y = \ln(\sin x) \) and its derivative \( \cot x \) using a calculator or graphing tool to confirm this result.

Key concepts

Derivative

A derivative, in calculus, represents the rate of change of a function concerning one of its variables. In simpler terms, it tells us how a function changes as its input changes. For a given function, the derivative can be thought of as the slope of the tangent line to the curve of the function at any point. Calculating derivatives is fundamental in calculus as it provides insights into the behavior of functions.

• The notation for a derivative of a function \( y \) with respect to \( x \) is \( \frac{dy}{dx} \).
• Understanding how to compute derivatives is essential for solving many problems in physics, engineering, and other sciences.

Recognizing and applying derivative rules, such as the product, quotient, and chain rules, is crucial for finding derivatives efficiently.

Chain Rule

The chain rule is a crucial technique in calculus for finding the derivative of a composition of functions. Simply put, it allows us to differentiate complex functions by breaking them down into simpler ones. If you have a function \( y = f(g(x)) \), then by using the chain rule, its derivative is calculated as \( \frac{dy}{dx} = f'(g(x)) \cdot g'(x) \).

• When you have one function nested inside another, the chain rule is the go-to method.
• It particularly shines when dealing with composite functions, like the one in this exercise, \( y = \ln(\sin x) \), which combines logarithmic and trigonometric functions.

Applying the chain rule makes it possible to break down the problem and handle each part separately, providing a structured approach to find complex derivatives effectively.

Trigonometric Functions

Trigonometric functions, such as \( \sin x \) and \( \cos x \), play a significant role in calculus, especially when dealing with derivatives of functions involving angles. These functions map angles to ratios of sides in right-angled triangles. In calculus:

• The sine function (\sin x) and its derivative being (\cos x) provide a foundation for solving many problems involving periodic or cyclic phenomena.

Such functions are fundamental due to their periodic nature and are widely used in modeling waves, oscillations, and other natural phenomena. They often appear together with other function types, necessitating the use of derivative rules.

Logarithmic Functions

Logarithmic functions, commonly denoted as \( \ln(x) \) in calculus, are the inverses of exponential functions. These functions grow very slowly compared to polynomials or exponentials, making them useful in various applications where natural scaling is required. A crucial aspect of logarithms in the context of derivatives is their ability to simplify multiplicative processes:

• The derivative of \( \ln(u) \) with respect to \( x \) is \( \frac{1}{u} \frac{du}{dx} \), which is a key feature when computing derivatives of more complex expressions.
• In the exercise provided, this property allows us to handle the composed function \( \ln(\sin x) \) by treating \( \sin x \) as \( u \) and differentiating accordingly.

Understanding logarithmic derivatives is essential for solving problems that involve growth, decay, and compounding, which occur across multiple scientific disciplines.