Problem 31 - Single Variable Calculus

Question

Find the derivative of the function. . \(y=\sqrt[3]{x}+\frac{1}{\sqrt{x}}\)

Step-by-Step Solution

Answer

The derivative of the given function \(y = \sqrt[3]{x} + \frac{1}{\sqrt{x}}\) is \(\frac{dy}{dx} = \frac{1}{3}x^{-\frac{2}{3}} - \frac{1}{2}x^{-\frac{3}{2}}\).

Step 1: Differentiate the first term: \(\frac{d}{dx}(x^{\frac{1}{3}})\)

Using the power rule, which states that \(\frac{d}{dx}(x^n) = nx^{n-1}\), we can calculate the derivative of the first term: \[\frac{d}{dx}(x^{\frac{1}{3}}) = \frac{1}{3}x^{\frac{1}{3} - 1} = \frac{1}{3}x^{-\frac{2}{3}}\]

Step 2: Differentiate the second term: \(\frac{d}{dx}(x^{-\frac{1}{2}})\)

Similarly, apply the power rule to differentiate the second term: \[\frac{d}{dx}(x^{-\frac{1}{2}}) = -\frac{1}{2}x^{-\frac{1}{2} - 1} = -\frac{1}{2}x^{-\frac{3}{2}}\]

Step 3: Combine the individual derivatives

The derivative of the entire function is the sum of the derivatives of each term: \[\frac{dy}{dx} = \frac{1}{3}x^{-\frac{2}{3}} - \frac{1}{2}x^{-\frac{3}{2}}\] This is the final answer for the derivative of the given function.

Key Concepts

Power Rule

The Power Rule is a fundamental technique in calculus for finding the derivative of power functions. Think of it as a shortcut to differentiate functions of the form \(x^n\). Using it makes complex-looking derivatives much more manageable.

The formula for the Power Rule is simple: \(\frac{d}{dx}(x^n) = nx^{n-1}\). Essentially, you bring the exponent down as a coefficient and then subtract one from the exponent.

Here's a quick breakdown:

• Identify the exponent \(n\) of the term you are differentiating.
• Multiply the term by \(n\).
• Decrease the original exponent by 1.

For instance, in our example, \(\frac{d}{dx}(x^{\frac{1}{3}})\) results in \(\frac{1}{3}x^{-\frac{2}{3}}\) because you multiply by \(\frac{1}{3}\) and subtract one from \(\frac{1}{3}\).

The same rule is applied to \(x^{-\frac{1}{2}}\) resulting in \(-\frac{1}{2}x^{-\frac{3}{2}}\). Once you get the hang of it, you'll see how effortless finding derivatives becomes.

Differentiation

Differentiation is the process of finding a derivative, which is pivotal in calculus. A derivative represents the rate of change or the slope of the function at any given point. It's like finding out how fast a car is going at an exact moment.

When you differentiate a function, you're essentially applying rules to systematically find this rate of change.

Let's break it down:

• The derivative gives you a new function that tells you how your original function is changing at every point.
• Differentiation uses several rules, like the Power Rule, to make this easier.
• In complex functions, like sums, each part is differentiated separately, and then combined.

In the original problem, differentiation was applied to each term: \(x^{\frac{1}{3}}\) and \(x^{-\frac{1}{2}}\). This involved using the Power Rule and combining the results. The final derivative tells us how \(y = \sqrt[3]{x} + \frac{1}{\sqrt{x}}\) behaves as its inputs change.

Calculus

Calculus is a vast field of mathematics that studies continuous change, and it's split mainly into differentiation and integration. The focus in our exercise is on differentiation, which helps us understand how functions change instantly.

Here are key points about calculus and differentiation:

• Calculus was developed to solve problems in physics, such as motion and change.
• Differentiation is like zooming in on a curve to see its exact slope at a specific point.
• Understanding the slope can help solve real-world problems like predicting future events based on current trends.

When we differentiate, we're applying calculus principles to break down complex problems, making them simpler. Calculus provides tools, like the Power Rule, to find derivatives easily and comprehend how functions behave. By mastering calculus, you'll have the ability to tackle a wide range of mathematical and scientific challenges.