교과서 질문
15–48. Derivatives Find the derivative of the following functions.
y = In (x³+1)^π
366
views
Ch. 3 - Derivatives
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
당신이 사용하는 게 아니라요?교과서 변경다른 교과서를 선택하세요
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 3.10.71
67–78. Derivatives of inverse functions Consider the following functions (on the given interval, if specified). Find the derivative of the inverse function.
f(x) = e^3x+1
검증된 단계별 안내1
First, understand that if y = f(x) = e^(3x+1), then the inverse function, denoted as f^(-1)(y), is the function that satisfies x = f^(-1)(y).
To find the derivative of the inverse function, we use the formula: (f^(-1))'(y) = 1 / f'(x), where x = f^(-1)(y).
Calculate the derivative of the original function f(x) = e^(3x+1). Using the chain rule, the derivative f'(x) = 3e^(3x+1).
Substitute f'(x) into the inverse derivative formula: (f^(-1))'(y) = 1 / (3e^(3x+1)).
Since y = e^(3x+1), solve for x in terms of y to express x = f^(-1)(y). Then substitute this expression for x back into the formula for (f^(-1))'(y) to find the derivative of the inverse function in terms of y.
비슷한 문제에 대한 검증된 영상 답변:
이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
3m도움이 되었나요?
주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Inverse Functions
An inverse function essentially reverses the effect of the original function. If a function f takes an input x and produces an output y, the inverse function f⁻¹ takes y back to x. Understanding how to find and work with inverse functions is crucial for solving problems involving derivatives of these functions.
추천 영상:
4:49
Inverse Cosine
Derivative of a Function
The derivative of a function measures how the function's output changes as its input changes. It is a fundamental concept in calculus that provides information about the slope of the function at any given point. For the function f(x) = e^(3x+1), finding its derivative is the first step in determining the derivative of its inverse.
추천 영상:
06:30Derivatives of Other Trig Functions
Derivative of Inverse Functions Formula
The derivative of an inverse function can be found using the formula (f⁻¹)'(y) = 1 / f'(x), where y = f(x). This relationship shows that the derivative of the inverse function at a point is the reciprocal of the derivative of the original function at the corresponding point. This concept is essential for calculating the derivative of the inverse function in the given problem.
추천 영상:
07:26Derivatives of Inverse Sine & Inverse Cosine
관련 실천
교과서 질문
Derivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of √x.
f(v) = v¹⁰⁰+e^v+10
277
views
교과서 질문
Derivative calculations Evaluate the derivative of the following functions at the given point.
f(s) = 2√s-1; a=25
219
views
교과서 질문
Simplify the expression e^xln(x²+1).
287
views
교과서 질문
Tangent lines Find an equation of the line tangent to the graph of f at the given point.
f(x) = sec−1(ex); (ln 2,π/3)
252
views
교과서 질문
23–51. Calculating derivatives Find the derivative of the following functions.
y = a sin x + b cos x/a sin x - b cos x; a and b are nonzero constants
309
views