Ch. 3 - Derivatives

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 3 - Derivatives

Problem 44a

Chapter 3, Problem 44a

Use the definition of the derivative to determine d/dx (√ax+b), where a and b are constants.

Verified step by step guidance

Step 1: Recall the definition of the derivative, which is given by the limit: \( f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \).

Step 2: Identify the function \( f(x) = \sqrt{ax + b} \) and substitute it into the definition of the derivative.

Step 3: Compute \( f(x+h) = \sqrt{a(x+h) + b} = \sqrt{ax + ah + b} \).

Step 4: Substitute \( f(x) \) and \( f(x+h) \) into the derivative definition: \( f'(x) = \lim_{h \to 0} \frac{\sqrt{ax + ah + b} - \sqrt{ax + b}}{h} \).

Step 5: To simplify the expression, multiply the numerator and the denominator by the conjugate of the numerator: \( \frac{\sqrt{ax + ah + b} - \sqrt{ax + b}}{h} \times \frac{\sqrt{ax + ah + b} + \sqrt{ax + b}}{\sqrt{ax + ah + b} + \sqrt{ax + b}} \).

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Key Concepts

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

Definition of the Derivative

The derivative of a function at a point measures the rate at which the function's value changes as its input changes. Formally, it is defined as the limit of the average rate of change of the function as the interval approaches zero. This concept is foundational in calculus, as it provides a way to understand how functions behave locally.

Limit Process

The limit process is a fundamental concept in calculus that allows us to evaluate the behavior of functions as they approach a certain point. In the context of derivatives, it involves taking the limit of the difference quotient, which is the ratio of the change in the function's value to the change in the input, as the change in input approaches zero. This process is crucial for defining derivatives accurately.

Chain Rule

The chain rule is a formula for computing the derivative of the composition of two or more functions. It states that if you have a function that is the composition of two functions, the derivative can be found by multiplying the derivative of the outer function by the derivative of the inner function. This rule is particularly useful when differentiating functions like √(ax + b), where the inner function is ax + b and the outer function is the square root.

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