91Ó°ÊÓ

The power rule is a fundamental principle in calculus, used for finding the derivative of functions of the form \( y = x^n \). It states that if you have a function \( y = x^n \), the derivative of the function with respect to \( x \) is given by \( \frac{dy}{dx} = n x^{n-1} \). This means we multiply the original exponent by the coefficient and then reduce the exponent by one.
For example, if \( y = x^{3} \), according to the power rule, the derivative would be \( \frac{d y}{d x} = 3 x^{2} \).
In the exercise provided, we have \( y = x^{0.7} \). Using the power rule, we substitute \( n \) with \( 0.7 \) to get the derivative.

The derivative of a function indicates how the function's output value changes as the input value changes. It is a measure of the rate of change or how a quantity reacts over time. The notation \( \frac{d y}{d x} \) represents the derivative of \( y \) with respect to \( x \).
In the given exercise, to find the derivative of the function \( y = x^{0.7} \), we apply the power rule. By performing the differentiation, we get:
\[ \frac{d y}{d x} = 0.7 x^{0.7-1} \]
Which simplifies to \[ \frac{d y}{d x} = 0.7 x^{-0.3} \]
Remember that differentiation helps us understand how a function behaves and how sensitive it is to changes in its input.

Simplifying an expression is an important step in ensuring your final answer is as clear and concise as possible. After applying the power rule to find the derivative in the given exercise, we obtain the expression \[ \frac{d y}{d x} = 0.7 x^{-0.3} \].
This expression is already in a simplified form. Simplifying an expression means reducing it to its most basic elements without changing its value.
In some cases, you might need to rewrite the derivative in a different form or factor it for better understanding. Here, it is essential to understand that \( x^{-0.3} \) indicates \( x \) raised to the power of \(-0.3 \), which is the same as \( \frac{1}{x^{0.3}} \). However, both forms are correct and can be used interchangeably.