y=xsin^-1x + square root of 1-x^2
and the question is ...?
y' = sin^-1(x)
Find the derivative of y with respect to the appropriate variable
The given expression is y = x * sin^(-1)(x) + √(1 - x^2).
To understand how to evaluate this expression, let's break it down step by step:
1. The first term is x * sin^(-1)(x), where sin^(-1)(x) represents the inverse sine function (also known as arcsine function). The inverse sine function returns the angle whose sine is x. Since this function involves trigonometry, the angle is usually measured in radians.
2. The second term is √(1 - x^2), which represents the square root of (1 - x^2). This is a standard mathematical operation to find the square root of a number.
To evaluate this expression for a given value of x, follow these steps:
Step 1: Calculate the inverse sine function (sin^(-1)(x))
- Start by finding the angle (in radians) whose sine is x. Remember that the range of the inverse sine function is -π/2 to π/2 radians.
- Use a trigonometric function calculator or lookup table to determine the angle corresponding to x.
Step 2: Substitute the value of sin^(-1)(x) in the expression
- Multiply x by the angle obtained in step 1.
- Take the square root of (1 - x^2).
Step 3: Add the two terms together to get the final value of y.
Keep in mind that this expression may have restrictions on the values of x. For instance, the square root term (√(1 - x^2)) requires that the value inside the square root is non-negative. So, the expression is valid only when 1 - x^2 ≥ 0.
By following these steps, you can evaluate the given expression for any value of x within the permissible domain.