Domanda del libro di testo
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
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Capitolo 3, Problema 33
Calculate the derivative of the following functions.
y = sec(3x+1)
Guida verificata passo dopo passo1
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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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.
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Percorso guidato
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.
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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.
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Percorso guidato
6:22Graphs of Secant and Cosecant Functions
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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.
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Equations of tangent lines by definition (2)
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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.
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Domanda del libro di testo
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.
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