교과서 질문
Derivatives of inverse functions from a table Use the following tables to determine the indicated derivatives or state that the derivative cannot be determined. <IMAGE>
b. (f^-1)'(6)
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Ch. 3 - Derivatives
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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TOC.tableOfContents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
3장, 문제 3.2.29b
21–30. Derivatives
b. Evaluate f'(a) for the given values of a.
f(s) = 4s³+3s; a= -3, -1
검증된 단계별 안내1
Step 1: Identify the function f(s) = 4s^3 + 3s and the values of a for which we need to evaluate the derivative, which are a = -3 and a = -1.
Step 2: Find the derivative of the function f(s) with respect to s. Use the power rule, which states that the derivative of s^n is n*s^(n-1).
Step 3: Apply the power rule to each term in the function. The derivative of 4s^3 is 12s^2, and the derivative of 3s is 3.
Step 4: Combine the derivatives to get the derivative function f'(s) = 12s^2 + 3.
Step 5: Evaluate the derivative at the given values of a. Substitute s = -3 into f'(s) to find f'(-3), and substitute s = -1 into f'(s) to find f'(-1).
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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 any given point. The derivative can be computed using various rules, such as the power rule, product rule, and quotient rule.
추천 영상:
가이드 코스
05:44
Derivatives
Power Rule
The power rule is a basic differentiation technique used to find the derivative of functions in the form f(s) = s^n, where n is a real number. According to this rule, the derivative f'(s) is calculated as n * s^(n-1). This rule simplifies the process of differentiation for polynomial functions, making it easier to evaluate derivatives at specific points.
추천 영상:
5:50Power Rules
Evaluating Derivatives at Specific Points
Evaluating the derivative at specific points involves substituting the given values into the derivative function. This process allows us to determine the instantaneous rate of change of the original function at those points. In the context of the question, we will first find the derivative of f(s) and then substitute a = -3 and a = -1 to find f'(-3) and f'(-1).
추천 영상:
04:50Critical Points
관련 실천
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Velocity of a car The graph shows the position s=f(t) of a car t hours after 5:00 P.M. relative to its starting point s=0,where s is measured in miles. <IMAGE>
b. At approximately what time is the car traveling the fastest? The slowest?
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교과서 질문
31–32. Velocity functions A projectile is fired vertically upward into the air, and its position (in feet) above the ground after t seconds is given by the function s(t).
b. Determine the instantaneous velocity of the projectile at t=1 and t = 2 seconds.
s(t)= −16t²+100t
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교과서 질문
13-26 Implicit differentiation Carry out the following steps.
b. Find the slope of the curve at the given point.
cos y = x; (0, π/2)
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교과서 질문
For what values of x does g(x) = x−sin x have a slope of 1?
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교과서 질문
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.
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