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Chapter 2: Problem 56

Finding a Derivative In Exercises \(43-64\) , find the derivative of the function. $$ g(\theta)=\cos ^{2} 8 \theta $$

Short Answer

Expert verified
The derivative of the function \(g(\theta)=\cos^2 8 \theta\) is \(-2\cos(8 \theta)\sin(8 \theta)\).

Step by step solution

01

Identify the Inner Function

The first step is to identify the 'inner' function in the given composition of functions. Here, the 'inner' function is \(\cos(8 \theta)\). Denote it as \(u\), so \(u = \cos(8 \theta)\).
02

Identify the Outer Function

The next step is to identify the 'outer' function in the composition. Here, the 'outer' function is \(x^2\), where \(x\) is the output of the 'inner' function. written in terms of \(u\), this is \(u^2\).
03

Apply the Chain Rule

To find the derivative of \(g(\theta)\), we can now apply the Chain Rule, which states that the derivative of a composition of functions is the derivative of the outer function evaluated at the inner function, times the derivative of the inner function. Using the derivatives established earlier, this would be \(2u * -\sin(u)\). Substituting cos(8鈩�) back in for u, the derivative is \(2\cos(8 \theta) * -\sin(8 \theta)\).
04

Simplify

Simplify the result to get the final derivative, obtaining \(-2\cos(8 \theta)\sin(8 \theta)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Chain Rule
The Chain Rule is a fundamental technique in calculus used to calculate the derivative of a composite function. Whenever you have a function that is composed of two or more functions, such as g(胃) = (肠辞蝉(8胃))虏 in our exercise, the Chain Rule allows you to differentiate it in a step by step process.

Put simply, if you have a function h(x) = f(g(x)), the derivative h'(x) is f'(g(x)) * g'(x). Here, 'g'(x)' represents the derivative of the inner function, while 'f'(g(x))' is the derivative of the outer function with the inner function plugged in. This technique is especially useful when the function's composition is not explicitly solvable or when direct differentiation is complex.
Derivative of Trigonometric Functions
Trigonometric functions are another cornerstone of calculus, and their derivatives are essential in many aspects of science and engineering. Each trigonometric function has a specific derivative. For example, the derivative of sin(x) is cos(x), and the derivative of cos(x) is -sin(x).

The knowledge of these derivatives is crucial when applying the Chain Rule, as we did in our exercise. After identifying the inner function, 肠辞蝉(8胃), and the outer function 虫虏, we used the fact that the derivative of cos(u) is -sin(u) to find the derivative of the composite function.
Function Composition
Function composition involves creating a new function by combining two or more functions. For instance, if f(x) = 虫虏 and g(x) = cos(x), then the composition f(g(x)), often written as f 鈭� g or (fog)(x), would be 肠辞蝉虏(虫).

In the exercise, the composed function g(胃) = cos虏(8胃) is the result of substituting the trigonometric function 肠辞蝉(8胃) into the power function 耻虏. Understanding composition is vital when identifying the 'inner' and 'outer' functions before applying the Chain Rule, which we then use to differentiate the composed function systematically.
Differentiation Techniques
Differentiation techniques are various methods used for finding the derivative of functions. These include the Power Rule, Product Rule, Quotient Rule, and Chain Rule, among others. Each technique is suitable for different kinds of functions or situations.

For single functions like 虫鈦�, the Power Rule is often used, dictating that the derivative is n虫鈦库伝鹿. For products of functions, the Product Rule is applied, and similarly, the Quotient Rule is used for ratios of functions. In the case of our exercise, the function was a composition of a trigonometric and a power function, which necessitated the use of the Chain Rule to find its derivative efficiently.

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Most popular questions from this chapter

Sketching a Graph Sketch the graph of a differentiable function \(f\) such that \(f(2)=0, f^{\prime} < 0\) for \(-\infty< x <2,\) and \(f^{\prime}>0\) for \(2< x <\infty\) . Explain how you found your answer.

Determining Differentiability In Exercises 89 and \(90,\) determine whether the function is differentiable at \(x=2\) . $$ f(x)=\left\\{\begin{array}{ll}{x^{2}+1,} & {x \leq 2} \\ {4 x-3,} & {x>2}\end{array}\right. $$

A ladder 25 feet long is leaning against the wall of a house (see figure). The base of the ladder is pulled away from the wall at a rate of 2 feet per second. (a) How fast is the top of the ladder moving down the wall when its base is 7 feet, 15 feet, and 24 feet from the wall? (b) Consider the triangle formed by the side of the house, the ladder, and the ground. Find the rate at which the area of the triangle is changing when the base of the ladder is 7 feet from the wall. (c) Find the rate at which the angle between the ladder and the wall of the house is changing when the base of the ladder is 7 feet from the wall.

Determining Differentiability In Exercises 89 and \(90,\) determine whether the function is differentiable at \(x=2\) . $$ f(x)=\left\\{\begin{array}{ll}{\frac{1}{2} x+1,} & {x<2} \\ {\sqrt{2 x},} & {x \geq 2}\end{array}\right. $$

Using Relationships In Exercises \(103-106,\) use the given information to find \(f^{\prime}(2) .\) $$ \begin{array}{l}{g(2)=3 \quad \text { and } \quad g^{\prime}(2)=-2} \\\ {h(2)=-1 \quad \text { and } \quad h^{\prime}(2)=4}\end{array} $$ $$ f(x)=4-h(x) $$
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