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) = 1 + √x

Verified step by step guidance

Identify the function g(x) = 1 + √x and the point at which you want to find the derivative, which is x.

Substitute g(x) into the alternative formula for derivatives: g'(x) = lim (z → x) [(g(z) − g(x)) / (z − x)].

Express g(z) in terms of z: g(z) = 1 + √z. Substitute this into the formula: g'(x) = lim (z → x) [(1 + √z − (1 + √x)) / (z − x)].

Simplify the expression inside the limit: g'(x) = lim (z → x) [(√z − √x) / (z − x)].

To evaluate the limit, multiply the numerator and the denominator by the conjugate of the numerator: (√z + √x). This will help eliminate the square roots and simplify the expression for taking the limit.

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Key Concepts

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

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. Understanding derivatives is crucial for analyzing the behavior of functions, such as finding slopes of tangent lines and rates of change.

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, and they help in understanding the behavior of functions at points where they may not be explicitly defined. Calculating limits involves evaluating the function's behavior as the variable approaches a specific value.

Alternative Formula for Derivatives

The alternative formula for derivatives, f'(x) = lim (z → x) (f(z) − f(x)) / (z − x), provides a way to calculate the derivative using limits. This formula emphasizes the concept of instantaneous rate of change by considering the difference quotient as z approaches x. It is particularly useful for functions where the standard derivative rules are not easily applicable, requiring a deeper understanding of limits and function behavior.

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Derivatives

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Textbook Question

The general polynomial of degree n has the form

P(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₂x² + a₁x + a₀,

where aₙ ≠ 0. Find P'(x).

Textbook Question

In Exercises 11–18, find the slope of the function’s graph at the given point. Then find an equation for the line tangent to the graph there.

g(x) = 8 / x², (2, 2)