Domanda del libro di testo
Derivatives and tangent lines
a. For the following functions and values of a, find f′(a).
f(x) = x²; a=3
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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 34
Calculate the derivative of the following functions.
y = csc ex
Guida verificata passo dopo passo1
Step 1: Identify the function to differentiate. The function given is \( y = \csc(e^x) \).
Step 2: Recall the derivative of the cosecant function. The derivative of \( \csc(u) \) with respect to \( u \) is \( -\csc(u)\cot(u) \).
Step 3: Apply the chain rule. Since \( u = e^x \), differentiate \( u \) with respect to \( x \), which is \( \frac{du}{dx} = e^x \).
Step 4: Combine the results using the chain rule. The derivative of \( y = \csc(e^x) \) is \( \frac{dy}{dx} = -\csc(e^x)\cot(e^x) \cdot e^x \).
Step 5: Simplify the expression if possible. The derivative is \( \frac{dy}{dx} = -e^x \csc(e^x)\cot(e^x) \).
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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 a fundamental concept in calculus that provides the slope of the tangent line to the curve of the function at any given point. The derivative is often denoted as f'(x) or dy/dx, and it can be calculated using various rules such as the power rule, product rule, and chain rule.
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Percorso guidato
05:44
Derivatives
Cosecant Function (csc)
The cosecant function, denoted as csc(x), is the reciprocal of the sine function, defined as csc(x) = 1/sin(x). In the context of derivatives, understanding the properties and behavior of trigonometric functions like cosecant is essential, especially when applying differentiation rules. The derivative of csc(x) is -csc(x)cot(x), which is crucial for differentiating functions involving csc.
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Percorso guidato
6:22Graphs of Secant and Cosecant Functions
Chain Rule
The chain rule is a fundamental technique in calculus used to differentiate composite functions. It states that if a function y = f(g(x)) is composed of two functions, the derivative can be found by multiplying the derivative of the outer function f with the derivative of the inner function g. This rule is particularly important when dealing with functions like y = csc(e^x), where e^x is the inner function.
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05:02Intro to the Chain Rule
Pratica correlata
Domanda del libro di testo
Derivatives and tangent lines
b. Determine an equation of the line tangent to the graph of f at the point (a,f(a)) for the given value of a.
f(x) = x²; a=3
322
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Domanda del libro di testo
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?
582
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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.
For what prices is the demand elastic? Inelastic?
245
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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.
Find the elasticity function for this demand function.
248
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Domanda del libro di testo
Calculate the derivative of the following functions.
y = tan ex
405
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