Differentials Consider the following functions and express the relationship between a small change in x and the corresponding change in y in the form dy = f'(x)dx.


f(x) = sin⁻¹ x

Verified step by step guidance

1. First, understand that the problem is asking for the differential dy in terms of dx for the function f(x) = sin⁻¹(x). The differential dy represents the change in y corresponding to a small change in x, denoted as dx.
2. To find dy, we need to determine the derivative of the function f(x) = sin⁻¹(x) with respect to x. The derivative of sin⁻¹(x) is given by f'(x) = \( \frac{1}{\sqrt{1-x^2}} \).
3. Now, express the differential dy in terms of dx using the formula dy = f'(x)dx. Substitute the derivative we found into this formula.
4. This gives us dy = \( \frac{1}{\sqrt{1-x^2}} \) dx. This equation shows how a small change in x (dx) affects the change in y (dy) for the function f(x) = sin⁻¹(x).
5. Finally, remember that this relationship is valid for values of x within the domain of the inverse sine function, which is -1 ≤ x ≤ 1. This ensures that the expression under the square root is non-negative.

Key Concepts

Differentials

Differentials represent the infinitesimal changes in variables. In calculus, if y is a function of x, the differential dy is defined as the product of the derivative f'(x) and the differential dx, which represents a small change in x. This relationship helps in approximating how a small change in x affects the change in y.

Derivative

The derivative of a function at a point measures the rate at which the function's value changes as its input changes. It is denoted as f'(x) and is calculated as the limit of the average rate of change of the function as the interval approaches zero. In the context of differentials, the derivative provides the necessary slope to relate changes in x and y.

Inverse Function

An inverse function reverses the effect of the original function. For example, the inverse of f(x) = sin(x) is f⁻¹(x) = sin⁻¹(x), which takes a value from the range of the sine function and returns the corresponding angle. Understanding inverse functions is crucial when dealing with their derivatives, as the derivative of an inverse function can be found using the formula f⁻¹'(y) = 1 / f'(x) where y = f(x).