Using the Alternative Formula for Derivatives

Use the formula
f'(x) = lim (z → x) (f(z) − f(x)) / (z − x)
to find the derivative of the functions in Exercises 23–26.

g(x) = x / (x − 1)

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Identify the function g(x) = x / (x - 1) and the formula for the derivative: f'(x) = lim (z → x) (f(z) − f(x)) / (z − x).

Substitute g(x) into the formula: g'(x) = lim (z → x) ((z / (z - 1)) - (x / (x - 1))) / (z - x).

Find a common denominator for the fractions in the numerator: (z / (z - 1)) and (x / (x - 1)). The common denominator is (z - 1)(x - 1).

Rewrite the expression: g'(x) = lim (z → x) ((x(z - 1) - z(x - 1)) / ((z - 1)(x - 1))) / (z - x).

Simplify the expression in the numerator: xz - x - zx + z = z - x. Substitute back into the limit: g'(x) = lim (z → x) ((z - x) / ((z - 1)(x - 1)(z - x))). Cancel (z - x) and evaluate the limit as z approaches x.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Derivative

The derivative of a function at a point measures the rate at which the function's value changes as its input changes. It is a fundamental concept in calculus, representing the slope of the tangent line to the function's graph at a given point. Understanding derivatives is crucial for analyzing and predicting the behavior of functions.

Limit

A limit is a fundamental concept in calculus that describes the value a function approaches as the input approaches a certain point. Limits are essential for defining derivatives and integrals, as they allow us to handle values that are not easily computed directly. In this context, the limit is used to find the derivative by considering the behavior of the function as z approaches x.

Alternative Formula for Derivatives

The alternative formula for derivatives, f'(x) = lim (z → x) (f(z) − f(x)) / (z − x), provides a way to compute the derivative by considering the limit of the difference quotient as z approaches x. This formula is particularly useful for functions where the standard definition of the derivative might be cumbersome, allowing for a more intuitive approach to finding the rate of change at a point.

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