Question

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

Short answer

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

Step-by-step solution

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.

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 \( d(\sqrt{1-x^{2}}) = \frac{-x}{\sqrt{1-x^{2}}} dx \).

Key concepts

Chain Rule

Understanding the chain rule is essential when dealing with composite functions – 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=\