교과서 질문
Explain why b^x = e^xlnb.
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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 3.9.19
Find the derivative of the following functions.
y = In |sin x|
검증된 단계별 안내1
Step 1: Recognize that the function y = ln|sin(x)| is a composition of functions, specifically the natural logarithm function and the absolute value of the sine function. To find the derivative, we will use the chain rule.
Step 2: Apply the chain rule. The chain rule states that if you have a composite function y = f(g(x)), then the derivative y' = f'(g(x)) * g'(x). Here, f(u) = ln(u) and g(x) = |sin(x)|.
Step 3: Differentiate the outer function f(u) = ln(u) with respect to u. The derivative of ln(u) is 1/u. So, f'(u) = 1/u.
Step 4: Differentiate the inner function g(x) = |sin(x)| with respect to x. The derivative of |sin(x)| is (sin(x)/|sin(x)|) * cos(x), using the derivative of the absolute value function and the derivative of sin(x).
Step 5: Combine the results from steps 3 and 4 using the chain rule. The derivative of y = ln|sin(x)| is y' = (1/|sin(x)|) * (sin(x)/|sin(x)|) * cos(x). Simplify the expression to get the final derivative.
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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. Derivatives are fundamental in calculus for understanding rates of change and are used in various applications, including optimization and motion analysis.
추천 영상:
가이드 코스
05:44
Derivatives
Chain Rule
The chain rule is a formula for computing the derivative of the composition of two or more functions. It states that if you have a function that is the composition of two functions, the derivative can be found by multiplying the derivative of the outer function by the derivative of the inner function. This rule is essential when dealing with complex functions, such as logarithmic or trigonometric functions combined with others.
추천 영상:
05:02Intro to the Chain Rule
Logarithmic Differentiation
Logarithmic differentiation is a technique used to differentiate functions that are products or quotients of other functions, especially when they involve exponentials or logarithms. By taking the natural logarithm of both sides of an equation, one can simplify the differentiation process, making it easier to apply the product and chain rules. This method is particularly useful for functions like y = ln|sin x|, where the argument is a function itself.
추천 영상:
06:30Logarithmic Differentiation
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