Problem 37 - University Calculus: Early Transcendentals

Question

Find the derivatives of the function. $$y=\sqrt[7]{x^{2}}-x^{e}$$

Step-by-Step Solution

Answer

The derivative is \( \frac{dy}{dx} = \frac{2}{7}x^{-5/7} - ex^{e-1} \).

Step 1: Identify the Function Components

The function provided is \( y = \sqrt[7]{x^2} - x^e\). This can be expressed as \( y = x^{2/7} - x^e\), where \(e\) is the mathematical constant approximately equal to 2.718.

Step 2: Apply the Power Rule to Each Component

To differentiate the function, we'll apply the power rule to each term separately. The power rule states that if \( y = x^n \), then the derivative \( \frac{dy}{dx} = nx^{n-1}\).

Step 3: Differentiate \(x^{2/7}\)

Apply the power rule to find the derivative of \( x^{2/7} \). So, \( \frac{d}{dx}(x^{2/7}) = \frac{2}{7}x^{2/7-1} = \frac{2}{7}x^{-5/7} \).

Step 4: Differentiate \(x^e\)

Now, differentiate \( x^e \) using the power rule. Thus, \( \frac{d}{dx}(x^e) = ex^{e-1} \).

Step 5: Combine Derivatives

Combine the derivatives obtained in steps 3 and 4. The derivative of the function is \( \frac{dy}{dx} = \frac{2}{7}x^{-5/7} - ex^{e-1} \).

Key Concepts

Power Rule

The Power Rule is a fundamental tool in calculus, used to find the derivative of a function that has the form \( x^n \). When we talk about deriving functions, we are essentially finding how the function changes—the rate of change.

For a function \( y = x^n \), the derivative \( \frac{dy}{dx} \) is given by \( nx^{n-1} \). This rule is extremely helpful when dealing with powers of \( x \) because it simplifies the process significantly.

In our example function, the Power Rule is applied twice:

• To \( x^{2/7} \) where \( n = 2/7 \), the derivative becomes \( \frac{2}{7}x^{2/7-1} = \frac{2}{7}x^{-5/7} \).
• To \( x^e \) where \( n = e \), leading to the derivative \( ex^{e-1} \).

By using the Power Rule, we can easily address each term of the function separately and compute the derivative efficiently.

Constant e

The constant \( e \) is a unique and important number in mathematics, approximately equal to 2.718. It serves as the base for natural logarithms and arises naturally in the study of continuous growth processes.

In differentiation, \( e \) has special properties. A crucial one is that when \( e \) is used as an exponent or appears in functions like \( x^e \), its derivatives follow predictable patterns.

For the term \( x^e \) as part of our function, applying the Power Rule tells us the derivative is \( ex^{e-1} \). Here, \( e \) acts just like any constant exponent, but keep in mind that when differentiating \( e^x \), it retains its own value, adding to its significance in calculus.

Differentiation Techniques

Differentiation techniques allow us to compute derivatives which measure how a function changes as its input changes. Knowing how to differentiate correctly is essential for solving many problems in calculus.

The primary techniques include:

• The Power Rule, which simplifies differentiation of polynomial terms.
• The Constant Rule, where the derivative of a constant is always zero.
• Product, Quotient, and Chain Rules for more complex expressions.

For our function \( y = x^{2/7} - x^e \), the Power Rule was applied to each term independently, demonstrating how individual rules are used in tandem to solve differential calculus problems.

It's like having a toolkit—each technique serves its purpose, and by learning when and how to apply them, you can solve a broad array of mathematical challenges.