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The power rule is a basic but very important rule in differential calculus. It states that the derivative of a function of the form \(x^n\) is given by \(nx^{n-1}\). This rule simplifies the process of finding derivatives of polynomial functions by reducing the exponent by one and multiplying by the original exponent. For example, in our exercise, we needed to find the derivative of \(-x^{3/4}\). Applying the power rule, we get \(-\frac{3}{4}x^{-1/4}\). This significantly reduces the complexity of differentiating functions with polynomial forms.

Remember, always convert any roots or rational exponents into a simpler form that can be easily handled by the power rule.

Differential calculus is all about studying how functions change. It revolves around the concept of a derivative, which gives the rate at which a function is changing at any given point.

When you differentiate a function, you are finding its derivative. This function tells you how the original function behaves, how steep the curve is, and where it’s heading. Differential calculus provides powerful tools for solving real-world problems like finding speed, optimizing functions, or understanding growth rates.

The exercise we solved relies heavily on differential calculus. By differentiating the function, we observe how the rate of change is expressed, which is crucial in various applications like physics and engineering.

Simplifying derivatives is a key part of solving differential calculus problems. Once you have applied the relevant rule (like the power rule), you often get an expression that needs to be simplified to be fully useful.

In our case, after differentiating \(-x^{3/4}\), we obtained \(-\frac{3}{4}x^{-1/4}\). Simplifying this involves making the expression easier to understand and work with, typically by removing negative exponents or combining terms.

The final simplified form would be expressed as \(-\frac{3}{4} \frac{1}{x^{1/4}}\) or \(-\frac{3}{4x^{1/4}}\). Simplification ensures clarity and helps in further applications, such as in solving equations or analyzing functions.