Question

Differentiate the following functions: \(r=\sqrt{a \theta}\)

Short answer

Question: Find the derivative of the function \(r = \sqrt{a\theta}\) with respect to \(\theta\). Answer: \(\frac{dr}{d\theta} = \frac{a}{2\sqrt{a\theta}}\).

Step-by-step solution

Identify the function and its components

The given function is \(r = \sqrt{a\theta}\), where \(a\) is a constant. We can rewrite the function as \(r(\theta) = (\sqrt{a\theta})\), which shows that it's a function of \(\theta\). To differentiate \(r\) with respect to \(\theta\), we need to differentiate the square root function with respect to its argument, \(a\theta\).

Applying the chain rule

The chain rule states that if we have a function \(g(f(\theta))\), then the derivative of \(g\) with respect to \(\theta\) is \(g'(f(\theta)) * f'(\theta)\). In our case, we have \(g(f(\theta)) = \sqrt{a\theta}\), with \(g(x) = \sqrt{x}\) and \(f(\theta) = a\theta\). So we'll need to find the derivatives \(g'(x)\) and \(f'(\theta)\) and then apply the chain rule.

Differentiating the inner function \(f(\theta)\)

The inner function is \(f(\theta) = a\theta\). Since \(a\) is a constant, the derivative of \(a\theta\) with respect to \(\theta\) is simply \(a\). So, \(f'(\theta) = a\).

Differentiating the outer function \(g(x)\)

The outer function is \(g(x) = \sqrt{x}\). To differentiate it with respect to \(x\), we can rewrite it as \(g(x) = x^{\frac{1}{2}}\) and then use the power rule for differentiation. The power rule states that if \(g(x) = x^n\), then \(g'(x) = nx^{n-1}\). Applying the power rule, we have: \(g'(x) = \frac{1}{2}x^{\frac{1}{2}-1} = \frac{1}{2}x^{-\frac{1}{2}} = \frac{1}{2\sqrt{x}}\)

Applying the chain rule to find the derivative

Now that we have both \(g'(x)\) and \(f'(\theta)\), we can apply the chain rule to find the derivative of \(r(\theta)\) with respect to \(\theta\). According to the chain rule, we have: \(\frac{dr}{d\theta} = g'(f(\theta)) * f'(\theta) = \frac{1}{2\sqrt{a\theta}} * a = \frac{a}{2\sqrt{a\theta}}\) So the derivative of \(r\) with respect to \(\theta\) is: \(\frac{dr}{d\theta} = \frac{a}{2\sqrt{a\theta}}\).

Key concepts

Chain Rule

The chain rule is a fundamental concept in calculus, particularly useful when dealing with composite functions. In essence, it helps us find the derivative of a function that is composed of other functions. Here's how it works.

Suppose you have a function represented by two nested functions, like \(g(f(x))\). The chain rule states that to differentiate this composition, you need the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function. This can be written as:

• \( \frac{d}{dx}[g(f(x))] = g'(f(x)) \cdot f'(x) \).

For example, in the original exercise with \(r = \sqrt{a\theta}\), the chain rule is applied to differentiate this composite function. Here, \(g(x) = \sqrt{x}\) and \(f(\theta) = a\theta\). We find their derivatives and multiply them as as described to solve the problem.

So, the chain rule essentially serves as a bridge, allowing us to break down the layers of nested functions to effectively compute derivatives.

Power Rule

The power rule is a straightforward yet powerful tool in calculus for differentiating expressions of the form \((x^n)\). It simplifies the process of finding derivatives for functions that incorporate exponents. Let's take a closer look at how to use it.

According to the power rule, if a function \(g(x) = x^n\), where \(n\) is any real number, the derivative is given by:

• \( g'(x) = n \cdot x^{n-1} \).

In the context of the exercise, the power rule was applied to the equation \(g(x) = x^{\frac{1}{2}}\) to differentiate \((x^{\frac{1}{2}})\). By applying the power rule, we obtained: \(g'(x) = \frac{1}{2}x^{\frac{1}{2}-1} = \frac{1}{2}x^{-\frac{1}{2}} = \frac{1}{2\sqrt{x}}\).

This rule is extensively used because it's simple yet applicable in a wide range of problems, whether as part of the chain rule or independently.

Derivatives

Derivatives are a cornerstone of calculus and are essential in understanding how a function changes as its input changes. They provide a way to compute the rate of change or the slope of a function at any given point.

Derivatives are defined as a limit, and the derivative of a function \(f(x)\) with respect to \(x\) is essentially the limit of the average rate of change of the function over an interval as the interval approaches zero.

• \( f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \).

This concept is easy to visualize: imagine a tangent line touching a curve at a single point. The slope of this tangent line represents the derivative at that point. In the exercise conducted, finding the derivative of \(r = \sqrt{a\theta}\) involved using the chain rule as a method to handle composite functions.

Thus, derivatives are complex yet greatly valued in mathematics for their applications across various fields, such as physics, engineering, and economics.

Calculus

Calculus is an area of mathematics focusing on rates of change and accumulation. It includes two main branches: differential calculus, dealing with derivatives, and integral calculus, which focuses on integrals.

The fundamental idea of calculus is to study how quantities change and accumulate over time, making it essential for understanding real-world phenomena.

• **Differential Calculus**: Concentrates on derivatives as a way to find the rate at which things change. This part of calculus helps us analyze the behavior of functions, curves, and surfaces.


• **Integral Calculus**: Involves integration, which is about accumulation of quantities, such as areas under curves.

Calculus is built on foundational topics like limits, continuity, and infinite series.

In tackling the function \(r = \sqrt{a\theta}\), we engaged with differential calculus, using rules like the chain rule and power rule to find the rate of change of the function with respect to its variable. The result provides insight into how calculus molds our understanding of dynamic systems.