91Ó°ÊÓ

Calculus is a branch of mathematics that focuses on rates of change and the accumulation of quantities. It is divided into two main disciplines: differential calculus and integral calculus. Differential calculus is about understanding how things change. It deals with the concept of the derivative, which measures how a function changes as its input changes. The original problem of finding the derivative, or rate of change, of a function is a typical question you would encounter in a calculus class.

For the given function, $$f(x)=\frac{x^{4 / 3}}{4}$$, we needed to find \(f^{\backprime}(x)\). This involves applying the concepts of calculus, specifically differentiation, to determine the rate at which \(f(x)\) changes with respect to x.

The power rule is one of the basic techniques you'll learn early in differentiation. It makes finding derivatives straightforward when dealing with polynomial functions. The power rule states that if you have a function of the form \( f(x) = x^n \), its derivative is \( f^{\backprime}(x) = nx^{n-1} \). This rule works for any real number n, including fractions and negative numbers.

In the given problem, we started with the function \( f(x) = \frac{1}{4}x^{4/3} \). To apply the power rule, we identify n as 4/3. Using the power rule formula, we get:

• Multiply the exponent (4/3) by the coefficient (1/4)
• Subtract 1 from the exponent (4/3 - 1 = 1/3)

This leads us to: \( f^{\backprime}(x) = \frac{1}{4} \times \frac{4}{3} x^{(4/3 - 1)} \ = \frac{1}{3} x^{1/3} \).

Differentiation is the process of finding the derivative of a function. The derivative provides a measure of how a function's output value changes as its input value changes. This is essential for understanding various physical phenomena, from motion to growth rates.

Let's summarize the steps used in the differentiation of \( f(x) = \frac{x^{4 / 3}}{4} \):

• Rewrite the function for clarity: \( f(x) = \frac{1}{4}x^{4/3} \)
• Apply the power rule: \( f^{\backprime}(x) = \frac{1}{4} \times \frac{4}{3} x^{1/3} \)
• Simplification leads to: \( f^{\backprime}(x) = \frac{1}{3} x^{1/3} \)

Each step involves using fundamental calculus principles to manipulate the expression into a simplified derivative form. Understanding these steps will make solving similar problems much more manageable.