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Ch. 3 - Derivatives
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Briggs 3rd EditionCh. 3 - Derivatives문제 3.4.36
3장, 문제 3.4.36

Find and simplify the derivative of the following functions.
f(x) = ex(x3 − 3x2 + 6x − 6)

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1
Step 1: Identify the function f(x) = e^x (x^3 - 3x^2 + 6x - 6) as a product of two functions: u(x) = e^x and v(x) = x^3 - 3x^2 + 6x - 6.
Step 2: Use the product rule for differentiation, which states that if you have a function h(x) = u(x)v(x), then the derivative h'(x) = u'(x)v(x) + u(x)v'(x).
Step 3: Differentiate u(x) = e^x. The derivative u'(x) = e^x, since the derivative of e^x with respect to x is e^x.
Step 4: Differentiate v(x) = x^3 - 3x^2 + 6x - 6. Use the power rule for each term: v'(x) = 3x^2 - 6x + 6.
Step 5: Substitute u(x), u'(x), v(x), and v'(x) into the product rule formula: f'(x) = e^x (3x^2 - 6x + 6) + e^x (x^3 - 3x^2 + 6x - 6). Simplify the expression by factoring out e^x.

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주요 개념

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Derivative

The derivative of a function measures how the function's output value changes as its input value changes. It is defined as the limit of the average rate of change of the function over an interval as the interval approaches zero. In calculus, the derivative is often denoted as f'(x) or df/dx, and it provides critical information about the function's behavior, such as its slope and points of tangency.
추천 영상:
가이드 코스
05:44
Derivatives

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 product rule states that the derivative of their product is given by u'v + uv'. This rule is essential when differentiating functions that are products of simpler functions, such as the function f(x) in the given question.
추천 영상:
05:18
The Product Rule

Exponential Function

An exponential function is a mathematical function of the form f(x) = a * e^(bx), where e is Euler's number (approximately 2.71828), and a and b are constants. The derivative of an exponential function is unique because it is proportional to the function itself, meaning that d/dx[e^(bx)] = b * e^(bx). Understanding the properties of exponential functions is crucial for simplifying derivatives involving them, as seen in the function f(x) provided.
추천 영상:
6:13
Exponential Functions
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