교과서 질문
Use definition (1) (p. 133) to find the slope of the line tangent to the graph of f at P.
f(x) = x2 - 5; P(3,4)
269
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 14a
7–14. Find the derivative the following ways:
a. Using the Product Rule (Exercises 7–10) or the Quotient Rule (Exercises 11–14). Simplify your result.
y = x² - 2ax +a² / x-a, where a is a constant
검증된 단계별 안내1
Step 1: Identify the function as a quotient of two functions, where the numerator is \( u(x) = x^2 - 2ax + a^2 \) and the denominator is \( v(x) = x - a \).
Step 2: Recall the Quotient Rule for derivatives, which states that if \( y = \frac{u(x)}{v(x)} \), then \( y' = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2} \).
Step 3: Compute the derivative of the numerator, \( u'(x) = \frac{d}{dx}(x^2 - 2ax + a^2) = 2x - 2a \).
Step 4: Compute the derivative of the denominator, \( v'(x) = \frac{d}{dx}(x - a) = 1 \).
Step 5: Substitute \( u(x) \), \( u'(x) \), \( v(x) \), and \( v'(x) \) into the Quotient Rule formula and simplify the expression.
비슷한 문제에 대한 검증된 영상 답변:
이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
7m도움이 되었나요?
주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Product Rule
The Product Rule is a formula used to find the derivative of the product of two functions. If u(x) and v(x) are two differentiable functions, the derivative of their product is given by (u*v)' = u'v + uv'. This rule is essential when dealing with expressions where two functions are multiplied together, allowing for the correct application of differentiation.
추천 영상:
05:18
The Product Rule
Quotient Rule
The Quotient Rule is used to differentiate a function that is the quotient of two other functions. If u(x) and v(x) are differentiable functions, the derivative of their quotient is given by (u/v)' = (u'v - uv') / v². This rule is particularly important when the function is expressed as a fraction, ensuring that the differentiation accounts for both the numerator and denominator.
추천 영상:
06:43The Quotient Rule
Simplification of Derivatives
Simplification of derivatives involves reducing the expression obtained after differentiation to its simplest form. This may include factoring, canceling common terms, or combining like terms. Simplifying the derivative is crucial for clarity and ease of interpretation, especially when analyzing the behavior of the function or finding critical points.
추천 영상:
가이드 코스
05:44Derivatives
관련 실천
교과서 질문
Find f′(x) if f(x) = 15e^3x.
375
views
교과서 질문
Suppose the position of an object moving horizontally along a line after t seconds is given by the following functions s = f(t), where s is measured in feet, with s > 0 corresponding to positions right of the origin.
Determine the velocity and acceleration of the object at t = 1.
f(t) = t2 − 4t; 0 ≤ t ≤ 5
344
views
교과서 질문
5–24. For each of the following composite functions, find an inner function u=g(x) and an outer function y=f(u) such that y=f(g(x)). Then calculate dy/dx.
y = (3x+7)¹⁰
350
views
교과서 질문
The legs of an isosceles right triangle increase in length at a rate of 2 m/s.
c. At what rate is the length of the hypotenuse changing?
320
views
교과서 질문
7–14. Find the derivative the following ways:
a. Using the Product Rule (Exercises 7–10) or the Quotient Rule (Exercises 11–14). Simplify your result.
y = x² - a² / x-a, where a is a constant
359
views