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Chapter 4: Problem 27

In Exercises \(27-30,\) find the differential. \(d\left(\sqrt{1-x^{2}}\right)\)

Short Answer

Expert verified
The differential of the function \( \sqrt{1-x^{2}} \) is \( \frac{-x}{\sqrt{1-x^{2}}} dx \).

Step by step solution

01

Differentiate the Function

To find the differential of a function, you first need to find its derivative. The function given is \( \sqrt{1-x^{2}} \). We can rewrite this function in the form \( (1-x^{2})^{1/2} \) to simplify the differentiation process. Using the chain rule (\( df/dx = df/du * du/dx \)) where \( u = 1-x^2 \), we get the derivative as \( d\left(\sqrt{1-x^{2}}\right) = \frac{-x}{\sqrt{1-x^{2}}} dx \). The negative sign indicates that the function decreases as x increases.
02

Substitute \(dx\)

After getting the derivative of the function, we now make the differential by multiplying the derivative by \(dx\). Therefore, we substitute \(dx\) into the equation. Thus the differential \(d(\sqrt{1-x^{2}})\) becomes \( d(\sqrt{1-x^{2}}) = \frac{-x}{\sqrt{1-x^{2}}} dx \).

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Key Concepts

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

Chain Rule
Understanding the chain rule is essential when dealing with composite functions 鈥� functions made up of two or more simpler functions. In calculus, we often find ourselves needing to differentiate a function of a function - and that's where the chain rule comes into play.

Let's break down the chain rule in a simple way: If you have a function, say, h(x), which is composed of another function, g(u), where u = f(x), then the derivative of h with respect to x is given by the product of the derivative of h with respect to u and the derivative of u with respect to x. In mathematical terms, this is represented as: \[\frac{dh}{dx} = \frac{dh}{du} \times \frac{du}{dx}\]
When applied to our example, we see that u=\

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Most popular questions from this chapter

True or False If \(u\) and \(v\) are differentiable functions, then \(d(u v)=d u d v .\) Justify your answer.

Minting Coins A manufacturer contracts to mint coins for the federal government. The coins must weigh within 0.1\(\%\) of their ideal weight, so the volume must be within 0.1\(\%\) of the ideal volume. Assuming the thickness of the coins does not change, what is the percentage change in the volume of the coin that would result from a 0.1\(\%\) increase in the radius?

Geometric Mean The geometric mean of two positive numbers \(a\) and \(b\) is \(\sqrt{a b}\) . Show that for \(f(x)=1 / x\) on any interval \([a, b]\) of positive numbers, the value of \(c\) in the conclusion of the Mean Value Theorem is \(c=\sqrt{a b} .\)

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Upper Bounds Show that for any numbers \(a\) and \(b\) \(|\sin b-\sin a| \leq|b-a|\)
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