Problem 11 - Thomas Calculus

Question

In Exercises \(5-36,\) find the derivative of \(y\) with respect to \(x, t,\) or \(\theta,\) as appropriate. $$ y=\ln (\theta+1) $$

Step-by-Step Solution

Answer

\( \frac{dy}{d\theta} = \frac{1}{\theta + 1} \)

Step 1: Identify the given function

The given function is \( y = \ln(\theta + 1) \). Our goal is to find the derivative \( \frac{dy}{d\theta} \) with respect to \( \theta \).

Step 2: Recall the derivative rule for natural logarithms

The derivative of \( \ln(u) \) with respect to \( u \) is \( \frac{1}{u} \) provided \( u > 0 \). In our case, \( u = \theta + 1 \).

Step 3: Derivative of the inside function

Since \( u = \theta + 1 \), find \( \frac{du}{d\theta} \). The derivative \( \frac{d}{d\theta}(\theta + 1) = 1 \).

Step 4: Apply the chain rule

Using the chain rule, the derivative \( \frac{dy}{d\theta} = \frac{d}{d\theta}[\ln(\theta + 1)] = \frac{1}{\theta + 1} \cdot 1 = \frac{1}{\theta + 1} \).

Step 5: Conclusion

Thus, the derivative of \( y \) with respect to \( \theta \) is \( \frac{1}{\theta + 1} \).

Key Concepts

Natural Logarithm

The natural logarithm, often abbreviated as "ln," is a logarithm with the base of the constant e, where e is approximately equal to 2.71828. It is a special case in mathematics because it arises naturally in many contexts, such as exponential growth, compound interest, and in differentiating functions.
The natural logarithm is particularly useful in calculus since it simplifies the process of differentiating and integrating logarithmic functions. When finding the derivative of a function involving a natural logarithm, like \( y = \ln(u) \), the formula to keep in mind is:

• The derivative \( \frac{d}{du}[\ln(u)] = \frac{1}{u} \) for \( u > 0 \).

This rule shows that the derivative is simply the reciprocal of the argument of the logarithm, making it quite straightforward to apply. Understanding this basic property allows us to tackle more complex derivatives involving natural logarithms with ease.

Chain Rule

The chain rule is an essential technique in calculus used to differentiate composite functions, which are functions that are nested within other functions. The chain rule states that if you have a composite function \( y = f(g(x)) \), then its derivative with respect to x can be found by multiplying the derivative of the outer function by the derivative of the inner function:

• \( \frac{dy}{dx} = f'(g(x)) \cdot g'(x) \)

In the exercise, the function \( y = \ln(\theta + 1) \) is a composite function. Here, the natural logarithm is our outer function, and \( \theta + 1 \) is the inner function. Thus, when applying the chain rule:

• First, find the derivative of the outer function with respect to the inner function: \( \frac{d}{d\theta}[\ln(\theta + 1)] = \frac{1}{\theta + 1} \).
• Then, multiply by the derivative of the inner function: \( \frac{d}{d\theta}(\theta + 1) = 1 \).
• Thus, the complete derivative is \( \frac{1}{\theta + 1} \cdot 1 = \frac{1}{\theta + 1} \).

By using the chain rule, we can systematically find derivatives of more complex functions without needing to simplify them first.

Calculus

Calculus is a fundamental branch of mathematics that deals with continuous change. Two main areas of calculus are differential calculus and integral calculus. Differential calculus, which is the focus of this exercise, involves finding the rate of change of quantities, typically expressed by derivatives.
Derivatives are a tool for understanding how a function changes at any given point and are used extensively in physics, engineering, economics, and many other fields. The derivative provides critical information such as the slope of a curve at a particular point, and it helps in finding local minima and maxima, optimizing functions, and understanding motion dynamics.The process we used to find the derivative \( \frac{dy}{d\theta} = \frac{1}{\theta + 1} \) is a typical application of differential calculus. By leveraging natural logarithms and the chain rule, we're able to efficiently compute how the function \( y = \ln(\theta + 1) \) changes as \( \theta \) varies. Calculus thus equips us with the necessary tools to decipher the behavior of functions, providing a deeper insight into the patterns that govern real-world phenomena.