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.

f(x) = x² − 3x + 4

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First, identify the function f(x) given in the problem, which is f(x) = x² − 3x + 4.

Next, substitute f(x) into the alternative formula for derivatives: f'(x) = lim (z → x) (f(z) − f(x)) / (z − x).

Calculate f(z) by substituting z into the function: f(z) = z² − 3z + 4.

Substitute f(z) and f(x) into the formula: f'(x) = lim (z → x) ((z² − 3z + 4) − (x² − 3x + 4)) / (z − x).

Simplify the expression inside the limit: f'(x) = lim (z → x) ((z² − x²) − 3(z − x)) / (z − x). Factor the numerator and evaluate the limit as z approaches x.

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주요 개념

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Derivative

A derivative represents the rate at which a function is changing at any given point and is a fundamental concept in calculus. It is the limit of the average rate of change of the function over an interval as the interval approaches zero. In this context, the derivative of f(x) = x² − 3x + 4 will be calculated using the alternative formula for derivatives.

Limit

The limit is a fundamental concept in calculus that describes the value that a function approaches as the input approaches a certain point. In the alternative formula for derivatives, the limit is used to find the instantaneous rate of change by considering the behavior of the function as z approaches x. Understanding limits is crucial for applying this formula effectively.

Alternative Formula for Derivatives

The alternative formula for derivatives, f'(x) = lim (z → x) (f(z) − f(x)) / (z − x), provides a method to calculate the derivative by considering the limit of the difference quotient as z approaches x. This formula is particularly useful for understanding the derivative conceptually, as it directly relates to the definition of the derivative as the slope of the tangent line at a point.

Using the Alternative Formula for DerivativesUse the formulaf'(x) = lim (z → x) (f(z) − f(x)) / (z − x)to find the derivative of the functions in Exercises 23–26.f(x) = x² − 3x + 4