Ch. 3 - Derivatives

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 3 - Derivatives

Problem 3.5.80

Chapter 3, Problem 3.5.80

Using identities Use the identity sin 2x=2 sin x cos x sin 2 to find d/dx (sin 2x). Then use the identity cos 2x = cos² x−sin² x to express the derivative of sin 2x in terms of cos 2x.

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Start by using the identity \( \sin 2x = 2 \sin x \cos x \). This identity helps us express \( \sin 2x \) in terms of \( \sin x \) and \( \cos x \).

Differentiate \( \sin 2x = 2 \sin x \cos x \) with respect to \( x \). Use the product rule for differentiation, which states that \( \frac{d}{dx}(u \cdot v) = u'v + uv' \). Here, \( u = \sin x \) and \( v = \cos x \).

Apply the product rule: \( \frac{d}{dx}(2 \sin x \cos x) = 2(\cos x \cdot \cos x + \sin x \cdot (-\sin x)) \). Simplify this expression to find \( \frac{d}{dx}(\sin 2x) \).

Now, use the identity \( \cos 2x = \cos^2 x - \sin^2 x \) to express \( \cos^2 x \) and \( \sin^2 x \) in terms of \( \cos 2x \). Substitute these into the derivative expression obtained in the previous step.

Simplify the expression to express \( \frac{d}{dx}(\sin 2x) \) in terms of \( \cos 2x \). This involves algebraic manipulation using the identity \( \cos 2x = \cos^2 x - \sin^2 x \).

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Key Concepts

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

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. They are essential for simplifying expressions and solving equations in calculus. The identities sin 2x = 2 sin x cos x and cos 2x = cos² x - sin² x are examples that help relate different trigonometric functions, making it easier to differentiate or integrate them.

Differentiation

Differentiation is a fundamental concept in calculus that involves finding the derivative of a function, which represents the rate of change of the function with respect to its variable. In this context, we apply the chain rule and product rule to differentiate sin 2x using the identity provided. Understanding how to differentiate trigonometric functions is crucial for solving problems related to motion, optimization, and other applications.

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 g(x) that is composed with another function f(x), the derivative is found by multiplying the derivative of f with respect to g by the derivative of g with respect to x. This rule is particularly useful when differentiating functions like sin(2x), where the inner function (2x) affects the outer function (sin).

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Related Practice

Textbook Question

Derivatives Find and simplify the derivative of the following functions.

h(x) = (x−1)(2x²-1) / (x³-1)

Textbook Question

At all times, the length of the long leg of a right triangle is 3 times the length x of the short leg of the triangle. If the area of the triangle changes with respect to time t, find equations relating the area A to x and dA/dt to dx/dt.

Textbook Question

Calculate the derivative of the following functions. In some cases, it is useful to use the properties of logarithms to simplify the functions before computing f'(x).

f(x) = In(sec⁴x tan² x)

Textbook Question

Derivatives Find and simplify the derivative of the following functions.

f(t) = t⁵/³e^t

Textbook Question

Evaluate the derivative of the following functions.

f(x) = sin^-1 2x

Textbook Question

Calculate the derivative of the following functions. In some cases, it is useful to use the properties of logarithms to simplify the functions before computing f'(x).

f(x) = In(3x + 1)⁴