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Ch. 3 - Derivatives
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 3 - DerivativesProblem 3.2.27b
Chapter 3, Problem 3.2.27b

21–30. Derivatives
b. Evaluate f'(a) for the given values of a.
f(t) = 1/√t; a=9, 1/4

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1
Step 1: Identify the function f(t) = \(\frac{1}{\sqrt{t}\)} and recognize that you need to find its derivative, f'(t).
Step 2: Rewrite the function in a form that is easier to differentiate: f(t) = t^{-1/2}.
Step 3: Use the power rule for differentiation, which states that if f(t) = t^n, then f'(t) = n \(\cdot\) t^{n-1}. Apply this to f(t) = t^{-1/2}.
Step 4: Calculate the derivative: f'(t) = -\(\frac{1}{2}\) \(\cdot\) t^{-3/2}.
Step 5: Evaluate f'(t) at the given values of a. First, substitute a = 9 into f'(t) to find f'(9), and then substitute a = \(\frac{1}{4}\) into f'(t) to find f'(\(\frac{1}{4}\)).

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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 a fundamental concept in calculus that provides information about the slope of the tangent line to the curve of the function at a given point. The derivative can be computed using various rules, such as the power rule, product rule, and quotient rule, depending on the form of the function.
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Derivatives

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 is particularly useful when dealing with functions that involve square roots or other composite forms.
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Intro to the Chain Rule

Evaluating Derivatives at Specific Points

To evaluate the derivative at a specific point, you first need to find the derivative function and then substitute the given value into this function. This process allows you to determine the instantaneous rate of change of the original function at that particular point. In this case, you will compute f'(9) and f'(1/4) for the function f(t) = 1/√t.
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Related Practice
Textbook Question

13-26 Implicit differentiation Carry out the following steps.

b. Find the slope of the curve at the given point.

tan xy = x+y; (0,0)

Textbook Question

13-26 Implicit differentiation Carry out the following steps.

b. Find the slope of the curve at the given point.

³√x+³√y⁴ = 2;(1,1)

Textbook Question

{Use of Tech} Angle of elevation A small plane, moving at 70 m/s, flies horizontally on a line 400 meters directly above an observer. Let θ be the angle of elevation of the plane (see figure). <IMAGE>


b. Graph dθ/dx as a function of x and determine the point at which θ changes most rapidly.

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Textbook Question

A differential equation is an equation involving an unknown function and its derivatives. Consider the differential equation y′′(t)+y(t) = 0.

b. Show that y = B cos t satisfies the equation for any constant B.

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Textbook Question

{Use of Tech} Power and energy The total energy in megawatt-hr (MWh) used by a town is given by E(t) = 400t+2400/π sin πt/12, where t≥0 is measured in hours, with t=0 corresponding to noon.

b. At what time of day is the rate of energy consumption a maximum? What is the power at that time of day?

Textbook Question

109-112 {Use of Tech} Calculating limits The following limits are the derivatives of a composite function g at a point a.

b. Use the Chain Rule to find each limit. Verify your answer by using a calculator.

limx→2(x2−3)5−1x−2{\(\displaystyle\]\lim\)_{x\(\to\)2}}\(\frac{\left(x^2-3\right)^5-1}{x-2}\)