91Ó°ÊÓ

The difference quotient is a crucial concept when approximating the derivative of a function. Essentially, it measures the average rate of change of the function over a small interval. For a function \( f(x) \), the difference quotient is expressed as:

• \( \frac{f(x+h) - f(x)}{h} \)

Here, \( h \) is a small increment in \( x \), and by making \( h \) very small, the difference quotient approaches the derivative of the function at that point.
For the given function \( f(x) = x^3 - 3x + 1 \), we applied this by considering \( f(1+h) - f(1) \). This led to simplifications resulting in \( h^2 + 3h \).
As \( h \) approaches zero, the approximation becomes more accurate, which is an essential aspect of calculus. This approximation helps students get a practical feel of how derivatives function at specific points.

The process of finding the limit of a function as a variable approaches a particular value is called limit evaluation. It is the backbone of finding derivatives in calculus. After simplifying the difference quotient expression to \( h^2 + 3h \), the next step was limit evaluation:

• We consider the limit as \( h \to 0 \).

For the expression \( h^2 + 3h \), plugging in \( h = 0 \) gives 0. Thus, limit evaluation removes the dependence on \( h \) and ensures what remains is the exact derivative value. This step is necessary because the definition of the derivative is the limit of the difference quotient as \( h \) approaches zero.

The power rule is a quick and efficient method for finding the derivative of functions with terms like \( x^n \). The rule states that:

• If \( f(x) = x^n \), then \( f'(x) = nx^{n-1} \).

This was applied to differentiate the given polynomial function \( f(x) = x^3 - 3x + 1 \). By the power rule:

• The derivative of \( x^3 \) is \( 3x^2 \).
• For \(-3x\), treating it as \(-3x^1\), the derivative is \(-3 \cdot 1x^{1-1} = -3 \).

The power rule simplifies this process significantly, allowing to quickly find \( f'(x) = 3x^2 - 3 \). Calculating the derivative at \( x = 1 \) confirmed our limit evaluation, showing that both approaches yield \( f'(1) = 0 \). This comprehensive agreement highlights the power rule as a reliable tool for derivative calculations.