교과서 질문
Given the function f and the point Q, find all points P on the graph of f such that the line tangent to f at P passes through Q. Check your work by graphing f and the tangent lines.
f(x) = 1/x; Q (-2, 4)
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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 68
Find f′(x), f′′(x), and f′′′(x) for the following functions.
f(x) = 3x3 + 5x2 + 6x
검증된 단계별 안내1
To find the first derivative, f′(x), apply the power rule to each term of the function f(x) = 3x^3 + 5x^2 + 6x. The power rule states that d/dx [x^n] = n*x^(n-1).
For the term 3x^3, apply the power rule: the derivative is 3 * 3 * x^(3-1) = 9x^2.
For the term 5x^2, apply the power rule: the derivative is 5 * 2 * x^(2-1) = 10x.
For the term 6x, apply the power rule: the derivative is 6 * 1 * x^(1-1) = 6.
Combine the derivatives of each term to get f′(x) = 9x^2 + 10x + 6. To find f′′(x) and f′′′(x), repeat the differentiation process on f′(x) using the power rule.
비슷한 문제에 대한 검증된 영상 답변:
이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
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주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Differentiation
Differentiation is the process of finding the derivative of a function, which represents the rate of change of the function with respect to its variable. The first derivative, f'(x), indicates the slope of the tangent line to the curve at any point, while higher-order derivatives, such as f''(x) and f'''(x), provide information about the curvature and acceleration of the function.
추천 영상:
가이드 코스
05:53
Finding Differentials
Power Rule
The Power Rule is a fundamental technique in calculus used to differentiate functions of the form f(x) = ax^n, where a is a constant and n is a real number. According to this rule, the derivative f'(x) is calculated as f'(x) = n * ax^(n-1). This rule simplifies the differentiation process, especially for polynomial functions like the one given in the question.
추천 영상:
5:50Power Rules
Higher-Order Derivatives
Higher-order derivatives are derivatives of derivatives, providing deeper insights into the behavior of a function. The second derivative, f''(x), indicates the concavity of the function, while the third derivative, f'''(x), can reveal information about the rate of change of the curvature. Understanding these derivatives is essential for analyzing the function's behavior beyond just its slope.
추천 영상:
02:42Higher Order Derivatives
관련 실천
교과서 질문
The following equations implicitly define one or more functions.
b. Solve the given equation for y to identify the implicitly defined functions y=f₁(x), y = f₂(x), ….
x+y³−xy=1 (Hint: Rewrite as y³−1=xy−x and then factor both sides.)
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교과서 질문
Let f(x) = 4√x - x.
Find all points on the graph of f at which the tangent line is horizontal.
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교과서 질문
The following equations implicitly define one or more functions.
a. Find dy/dx using implicit differentiation.
x+y³−xy=1 (Hint: Rewrite as y³−1=xy−x and then factor both sides.)
290
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교과서 질문
Let f(x) = 4√x - x.
Find all points on the graph of f at which the tangent line has slope -1/2.
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교과서 질문
Given the function f and the point Q, find all points P on the graph of f such that the line tangent to f at P passes through Q. Check your work by graphing f and the tangent lines.
f(x)=x²+1; Q(3, 6)
258
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