교과서 질문
Use definition (2) (p. 135) to find the slope of the line tangent to the graph of f at P.
f(x) = √(x + 3); P (1,2)
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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 33
Calculate the derivative of the following functions.
y = sec(3x+1)
검증된 단계별 안내1
Step 1: Identify the outer function and the inner function. Here, the outer function is \( \sec(u) \) and the inner function is \( u = 3x + 1 \).
Step 2: Recall the derivative of the secant function. The derivative of \( \sec(u) \) with respect to \( u \) is \( \sec(u) \tan(u) \).
Step 3: Apply the chain rule. The chain rule states that the derivative of a composite function \( y = f(g(x)) \) is \( f'(g(x)) \cdot g'(x) \).
Step 4: Differentiate the inner function \( u = 3x + 1 \) with respect to \( x \). The derivative \( \frac{du}{dx} \) is 3.
Step 5: Combine the results using the chain rule. The derivative of \( y = \sec(3x + 1) \) is \( \sec(3x + 1) \tan(3x + 1) \cdot 3 \).
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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 changes. It is a fundamental concept in calculus that represents the slope of the tangent line to the curve of the function at any given point. The derivative is denoted as f'(x) or dy/dx, and it can be calculated using various rules, such as the power rule, product rule, quotient rule, and chain rule.
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 y = f(g(x)), the derivative is given by dy/dx = f'(g(x)) * g'(x). This rule is essential when differentiating functions that are nested within each other, such as trigonometric functions with linear transformations.
추천 영상:
05:02Intro to the Chain Rule
Secant Function
The secant function, denoted as sec(x), is the reciprocal of the cosine function, defined as sec(x) = 1/cos(x). It is important in calculus, especially when dealing with derivatives of trigonometric functions. The derivative of sec(x) is sec(x)tan(x), and understanding this relationship is crucial for differentiating functions that involve secant.
추천 영상:
가이드 코스
6:22Graphs of Secant and Cosecant Functions
관련 실천
교과서 질문
Demand and elasticity Based on sales data over the past year, the owner of a DVD store devises the demand function , where D(p) is the number of DVDs that can be sold in one day at a price of p dollars.
a. According to the model, how many DVDs can be sold in a day at a price of \$10?
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교과서 질문
Equations of tangent lines by definition (2)
b. Determine an equation of the tangent line at P.
f(x) = √x+3; P (1,2)
325
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교과서 질문
Find and simplify the derivative of the following functions.
h(x) = (x − 1)(x3+ x2 + x+1)
1085
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교과서 질문
Based on sales data over the past year, the owner of a DVD store devises the demand function D(p) = 40 - 2p, where D(p) is the number of DVDs that can be sold in one day at a price of p dollars.
For what prices is the demand elastic? Inelastic?
245
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교과서 질문
Based on sales data over the past year, the owner of a DVD store devises the demand function D(p) = 40 - 2p, where D(p) is the number of DVDs that can be sold in one day at a price of p dollars.
Find the elasticity function for this demand function.
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