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The difference quotient is a critical concept in calculus and is essential for finding the derivative of a function. It provides a formula to calculate the slope of the tangent line to the curve of the function at any given point. By definition, the difference quotient for a function \( g(x) \) is expressed as:

• \( \frac{g(x + h) - g(x)}{h} \)

This formula is essentially a representation of the average rate of change of the function over an interval. As the interval, represented by \( h \), becomes extremely small (approaching zero), the difference quotient approaches the slope of the tangent line at a single point, which is the derivative.
For example, in the function \( g(x) = x^3 + 1 \), the application of the difference quotient involves substituting \( g(x + h) \) for the incremented function, and \( g(x) \) as the base function to create the formula \( \frac{(x + h)^3 + 1 - (x^3 + 1)}{h} \). This setup is the foundation for finding the derivative.

The binomial theorem is a powerful tool in mathematics used to expand expressions that are raised to a power, such as \((x + h)^3\). It simplifies the expansion process by using a combination of coefficients, variables, and powers. As a result, it helps in deriving lengthy polynomial expressions quickly and accurately.

• The expression \((x + h)^3\) is derived from applying the binomial theorem: \((x + h)^3 = x^3 + 3x^2h + 3xh^2 + h^3\).

This expansion assists in breaking down complex expressions necessary for solving difference quotients. By substituting this expanded form into the difference quotient formula, one can easily follow through with further simplifications and ultimately find the derivative more effectively.The repeated terms, such as \( 3x^2h \), highlight how each element's power increases systematically, which becomes crucial in calculating derivatives using polynomial forms.

The limit of a function is pivotal in calculus for defining the concept of a derivative. It allows us to analyze the behavior of functions as they approach a certain point. To find the derivative of \( g(x) = x^3 + 1 \), we take the limit of the difference quotient as \( h \) approaches zero:

• \( \lim_{h \to 0} \frac{g(x + h) - g(x)}{h} \).

The process simplifies through substitutions and cancellations based on polynomial expansions, like using the binomial theorem. When expanding and simplifying, all terms involving \( h \) will diminish as \( h \) approaches zero, except for those independent of \( h \).
Thus, as shown in the step-by-step solution, when the limit is applied to the simplified expression \( 3x^2 + 3xh + h^2 \), all terms including \( h \) disappear, resulting in the final derivative of \( g'(x) = 3x^2 \). This calculation reflects the instantaneous rate of change of the original function at any point \( x \). Understanding limits is essential to grasp derivatives fully, as it connects the difference quotient to a specific slope at a point.