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Chapter 3: Problem 16

Use Version I of the Chain Rule to calculate \(\frac{d y}{d x}\). $$y=\sin \frac{x}{4}$$

Short Answer

Expert verified
Answer: The derivative of \(y = \sin(\frac{x}{4})\) with respect to x is \(\frac{dy}{dx} = \frac{1}{4}\cos(\frac{x}{4})\).

Step by step solution

01

Identify the functions

In this case, we have an outer function, which is the sine function, and an inner function, which is x divided by 4. Let's denote these functions as follows: - For the outer function \(g(u) = \sin(u)\), here \(u\) is a placeholder for the inner function. - For the inner function \(f(x) = \frac{x}{4}\). Therefore, we can rewrite the given function as \(y = g(f(x))\).
02

Apply the Chain Rule formula

According to the Chain Rule, we need to take the derivative of the outer function and then the derivative of the inner function, multiplying them together: $$\frac{dy}{dx} = g'(f(x))\cdot f'(x)$$
03

Find the derivatives of the outer and inner functions

First, let's find the derivative of the outer function, g(u): $$g'(u) = \frac{d}{du}(\sin(u)) = \cos(u)$$ Now, let's find the derivative of the inner function, f(x): $$f'(x) = \frac{d}{dx}(\frac{x}{4}) = \frac{1}{4}$$
04

Substitute the derivatives into the Chain Rule formula

Now, we need to substitute the derivatives we found in step 3 into the Chain Rule formula: $$\frac{dy}{dx} = g'(f(x))\cdot f'(x) = \cos(f(x)) \cdot \frac{1}{4}$$
05

Replace f(x) with the original inner function

Finally, we need to replace f(x) in our derivative expression with the original inner function: $$\frac{dy}{dx} = \cos(\frac{x}{4}) \cdot \frac{1}{4}$$ Now we have found the derivative of the given function using the Chain Rule: $$\frac{dy}{dx} = \frac{1}{4}\cos(\frac{x}{4})$$

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Key Concepts

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

Derivative of Composite Functions
Understanding the derivative of composite functions is essential when dealing with functions within functions in calculus. The process involves applying the Chain Rule, a fundamental technique in differentiation. Imagine you're trying to find the rate at which two nested processes are changing at the same time. That's where the Chain Rule shines.

In our example, the composite function is \(y = \sin(\frac{x}{4})\). We identified the inner function, \(\frac{x}{4}\), and the outer function, \(\sin(u)\), where \(u\) is a placeholder for the inner function, leading to the representation \(y = g(f(x))\). The Chain Rule tells us to differentiate \(g(u)\) with respect to \(u\), and \(f(x)\) with respect to \(x\), and then multiply these derivatives together for the final result.
Calculus
Calculus is the mathematical study of continuous change, akin to the way geometry is the study of shape and algebra is the study of generalizations of arithmetic operations. It has two major branches, differential calculus and integral calculus, both of which are linked by the fundamental theorem of calculus.

In the realm of differential calculus, we focus on the concept of the derivative, which measures how a function changes as its input changes. It's like capturing a snapshot of a moment when you're analyzing how quickly a car accelerates over time. Our exercise delves into this branch by finding the derivative of a trigonometric function, specifically harnessing the power of the Chain Rule to compute how the output of the function changes as \(x\) varies.
Trigonometric Functions Derivative
Trigonometric functions, such as \(\sin(x)\), \(\cos(x)\), and \(\tan(x)\), are pervasive in mathematics and physics, describing phenomena like waves and periodic motion. Their derivatives are equally important and are commonly memorized in calculus for ease of differentiation.

The derivative of \(\sin(x)\) is \(\cos(x)\), and understanding this simple relationship is vital. When we encounter a trigonometric function within another function, knowing these derivatives allows us to apply the Chain Rule effectively, just like we see with \(\cos(\frac{x}{4})\) being the derivative of \(\sin(\frac{x}{4})\) in our exercise.
Differentiation Techniques
Differentiation is the process of finding the derivative of a function. There are several techniques to tackle various types of functions, such as using the Power Rule for polynomials, the Product Rule for products of functions, and the Quotient Rule for ratios of functions.

One of the more sophisticated techniques is the Chain Rule, used when differentiating composite functions. This rule is a powerful tool that breaks down complicated functions into simpler parts. In our example, we start by differentiating the inner function \(f(x) = \frac{x}{4}\) and the outer function \(g(u) = \sin(u)\) separately. Simplifying differentiation into these steps makes even the most complex equations manageable.

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Most popular questions from this chapter

Find the following higher-order derivatives. $$\left.\frac{d^{3}}{d x^{3}}\left(x^{4.2}\right)\right|_{x=1}$$

Prove the following identities and give the values of \(x\) for which they are true. $$\tan \left(2 \tan ^{-1} x\right)=\frac{2 x}{1-x^{2}}$$

Suppose \(f\) is differentiable on an interval containing \(a\) and \(b\), and let \(P(a, f(a))\) and \(Q(b, f(b))\) be distinct points on the graph of \(f\). Let \(c\) be the \(x\) -coordinate of the point at which the lines tangent to the curve at \(P\) and \(Q\) intersect, assuming that the tangent lines are not parallel (see figure). a. If \(f(x)=x^{2},\) show that \(c=(a+b) / 2,\) the arithmetic mean of \(a\) and \(b\), for real numbers \(a\) and \(b\) b. If \(f(x)=\sqrt{x}\), show that \(c=\sqrt{a b}\), the geometric mean of \(a\) and \(b\), for \(a > 0\) and \(b > 0\) c. If \(f(x)=1 / x,\) show that \(c=2 a b /(a+b),\) the harmonic mean of \(a\) and \(b,\) for \(a > 0\) and \(b > 0\) d. Find an expression for \(c\) in terms of \(a\) and \(b\) for any (differentiable) function \(f\) whenever \(c\) exists.

An observer is \(20 \mathrm{m}\) above the ground floor of a large hotel atrium looking at a glass-enclosed elevator shaft that is \(20 \mathrm{m}\) horizontally from the observer (see figure). The angle of elevation of the elevator is the angle that the observer's line of sight makes with the horizontal (it may be positive or negative). Assuming that the elevator rises at a rate of \(5 \mathrm{m} / \mathrm{s}\), what is the rate of change of the angle of elevation when the elevator is \(10 \mathrm{m}\) above the ground? When the elevator is \(40 \mathrm{m}\) above the ground?

General logarithmic and exponential derivatives Compute the following derivatives. Use logarithmic differentiation where appropriate. $$\frac{d}{d x}\left(x^{\cos x}\right)$$
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