Problem 10 - Calculus of a Single Variable

Question

In Exercises 3–24, use the rules of differentiation to find the derivative of the function. $$ g(x)=\sqrt[4]{x} $$

Step-by-Step Solution

Answer

The derivative of the function \(g(x)=\sqrt[4]{x}\) is \(g'(x)= \frac{1}{4} x^{-\frac{3}{4}}\) or \(g'(x)= \frac{1}{4\sqrt[4]{x^3}}\).

Step 1: Rewrite \(\sqrt[4]{x}\) as \(x^{\frac{1}{4}}\)

Firstly, convert the root notation to exponent notation. The 4th root of x, \(\sqrt[4]{x}\), can be rewritten as \(x^{\frac{1}{4}}\). So, \(g(x)=x^{\frac{1}{4}}\).

Step 2: Apply the Power Rule

Next, apply the power rule for differentiation, which states that if \(f(x) = x^n\), then the derivative \(f'(x) = n \cdot x^{n-1}\). Applying the power rule to our function \(g(x)=x^{\frac{1}{4}}\) gives \(g'(x)= \frac{1}{4} x^{\frac{1}{4}-1}\).

Step 3: Simplify the Result

Now, simplify to attain the final answer. The exponent in \(g'(x)= \frac{1}{4} x^{\frac{1}{4}-1}\) simplifies to \(-\frac{3}{4}\). Thus, the final answer for the derivative of \(g(x)=x^{\frac{1}{4}}\) is \(g'(x)= \frac{1}{4} x^{-\frac{3}{4}}\). Optionally, it can be written with a root in the denominator, as \(g'(x)= \frac{1}{4\sqrt[4]{x^3}}\).

Key Concepts

Power Rule

The power rule is an essential tool in calculus, particularly in differentiation. It allows us to find the derivative of functions that are powers of a variable. The rule states:

• If you have a function of the form \( f(x) = x^n \),
• then the derivative, \( f'(x) \), will be \( n \cdot x^{n-1} \).

This rule makes finding derivatives straightforward by simply multiplying the current exponent by the coefficient, and then reducing the exponent by one.

For example, with \( g(x) = x^{\frac{1}{4}} \), apply the power rule like this:

• Multiply the exponent \( \frac{1}{4} \) with the coefficient (which is implicitly 1, since it's not normally written),
• Resulting in a new coefficient of \( \frac{1}{4} \).
• Next, reduce the exponent by 1, leading to the new exponent \( \frac{1}{4} - 1 = -\frac{3}{4} \).

Thus, the derivative is \( g'(x) = \frac{1}{4} x^{-\frac{3}{4}} \). Such simple steps thanks to the power rule!

Derivative

In mathematics, the derivative represents how a function changes as its input changes. It is a measure of the function's rate of change or its sensitivity to the variable it's defined upon. Differentiation is the process of finding a derivative.

The derivative of a function \( g(x) \) is denoted as \( g'(x) \) or \( \frac{dg}{dx} \). When we differentiate, we are finding how \( y \) (the function) changes when \( x \) (the variable) changes infinitesimally.

• It provides insight into the behavior of functions, such as increasing or decreasing.
• Derivatives can tell us about the slope of the function at any point.

When you apply the power rule to \( g(x) = x^{\frac{1}{4}} \), you find \( g'(x) = \frac{1}{4} x^{-\frac{3}{4}} \). This tells us the slope of \( g(x) \) at any point \( x \). A negative exponent in the derivative suggests a different kind of slope behavior compared to a typical positive power.

Understanding derivatives is crucial in fields such as physics, engineering, and economics, where they are used to model real-world phenomena involving rates of change.

Exponent Notation

Exponent notation is a mathematical way to denote the power to which a number is raised. It is a shorthand for repeated multiplication, offering a more efficient and compact way to represent numbers and operations involving them.

In the expression \( x^n \),

• \( x \) is the base,
• \( n \) is the exponent or power.

The expression means \( x \) is multiplied by itself \( n \) times. For example, \( x^3 = x \cdot x \cdot x \).

Exponent notation can represent roots as well. For example, \( \sqrt[4]{x} \) can be rewritten as \( x^{\frac{1}{4}} \). This transformation, known as converting from root notation to exponent notation, simplifies the process of performing operations like differentiation.

When we have these fractional exponents, it's easy to apply rules like the power rule, allowing us to work with derivatives effectively. In the exercise, converting \( \sqrt[4]{x} \) to \( x^{\frac{1}{4}} \) was the first step, simplifying the differentiation process into straightforward operations.