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 3.2.25a

Chapter 3, Problem 3.2.25a

21–30. Derivatives
a. Use limits to find the derivative function f' for the following functions f.
f(x) = 1/x+1; a = -1/2;5

Verified step by step guidance

Step 1: Understand the problem. We need to find the derivative of the function f(x) = \(\frac{1}{x+1}\) using the definition of the derivative, which involves limits.

Step 2: Recall the definition of the derivative. The derivative f'(x) of a function f(x) at a point x is given by the limit: f'(x) = \(\lim\)_{h \(\to\) 0} \(\frac{f(x+h) - f(x)}{h}\).

Step 3: Substitute f(x) = \(\frac{1}{x+1}\) into the definition. This gives us: f'(x) = \(\lim\)_{h \(\to\) 0} \(\frac{\frac{1}{x+h+1}\) - \(\frac{1}{x+1}\)}{h}.

Step 4: Simplify the expression inside the limit. Find a common denominator for the fractions in the numerator: \(\frac{1}{x+h+1}\) - \(\frac{1}{x+1}\) = \(\frac{(x+1) - (x+h+1)}{(x+h+1)(x+1)}\) = \(\frac{-h}{(x+h+1)(x+1)}\).

Step 5: Substitute the simplified expression back into the limit and simplify further: f'(x) = \(\lim\)_{h \(\to\) 0} \(\frac{-h}{h(x+h+1)(x+1)}\). Cancel the h in the numerator and denominator, then evaluate the limit as h approaches 0.

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Key Concepts

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

Derivatives

A derivative represents the rate of change of a function with respect to its variable. It is defined as the limit of the average rate of change of the function as the interval approaches zero. In calculus, the derivative is often denoted as f'(x) and can be interpreted as the slope of the tangent line to the graph of the function at a given point.

Limits

Limits are a fundamental concept in calculus that describe the behavior of a function as its input approaches a certain value. They are essential for defining derivatives, as the derivative is calculated using the limit of the difference quotient. Understanding limits allows us to analyze functions at points where they may not be explicitly defined or where they exhibit discontinuities.

Difference Quotient

The difference quotient is a formula used to calculate the average rate of change of a function over an interval. It is expressed as (f(x+h) - f(x))/h, where h is the change in x. As h approaches zero, the difference quotient approaches the derivative of the function, providing a way to find instantaneous rates of change.

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Related Practice

Textbook Question

13-26 Implicit differentiation Carry out the following steps.

a. Use implicit differentiation to find dy/dx.

cos y = x; (0, π/2)

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Textbook Question

45–50. Tangent lines Carry out the following steps. <IMAGE>

a. Verify that the given point lies on the curve.

x³+y³=2xy; (1, 1)

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Textbook Question

Derivatives and tangent lines

a. For the following functions and values of a, find f′(a).

f(x) = √2x+1; a= 4

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Textbook Question

Tracking a dive A biologist standing at the bottom of an 80-foot vertical cliff watches a peregrine falcon dive from the top of the cliff at a 45° angle from the horizontal (see figure). <IMAGE>

a. Express the angle of elevation θ from the biologist to the falcon as a function of the height h of the bird above the ground. (Hint: The vertical distance between the top of the cliff and the falcon is 80−h.)

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Textbook Question

21–30. Derivatives

a. Use limits to find the derivative function f' for the following functions f.

f(t) = 3t⁴; a= -2, 2

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Textbook Question

Use the graph of f in the figure to do the following. <IMAGE>

a. Find the values of x in (-2,2) at which f is not continuous.

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