Ch. 3 - Derivatives

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 3 - Derivatives

Problem 34

Chapter 3, Problem 34

Calculate the derivative of the following functions.
y = csc e^x

Verified step by step guidance

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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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.

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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Intro to the Chain Rule

Related Practice

Textbook Question

Derivatives and tangent lines

a. For the following functions and values of a, find f′(a).

f(x) = x²; a=3

Textbook Question

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

Textbook Question

Demand and elasticity Based on sales data over the past year, the owner of a DVD store devises the demand function $D (p) = 40 - 2 p$ , 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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Textbook Question

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?

Textbook Question

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.

Textbook Question

Calculate the derivative of the following functions.

y = tan e^x

Calculate the derivative of the following functions.y = csc ex

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Key Concepts

Derivative

Cosecant Function (csc)

Chain Rule

Calculate the derivative of the following functions.
y = csc e^x