Problem 75
Question
$$ y=x-\sqrt{1-x^{2}} \sin ^{-1} x $$
Step-by-Step Solution
Verified Answer
The given function \(y(x) = x - \sqrt{1-x^2}\sin^{-1}x\) is a combination of a linear function (x), a square root function (\(\sqrt{1-x^2}\)), and an inverse trigonometric function (\(\sin^{-1}x\)). To construct the given function, the square root function is first subtracted from the linear function and then the inverse trigonometric function is multiplied with the square root part only. This results in the final expression: \( y(x) = x - \sqrt{1-x^2}\sin^{-1}x \).
1Step 1: Understand the function
The function is given by:
\( y(x) = x - \sqrt{1-x^2}\sin^{-1}x \)
It is a combination of a linear function (x), a square root function (\(\sqrt{1-x^2}\)), and an inverse trigonometric function (\(\sin^{-1}x\)).
2Step 2: Identify the individual components
Let's break down the function into its components:
1. Linear function: y(x) = x
2. Square root function: y(x) = \(\sqrt{1-x^2}\)
3. Inverse trigonometric function: y(x) = \(\sin^{-1}x\)
Now we are going to combine these components in sequence to construct the given function.
3Step 3: Combine the linear function and square root function
Subtracting the square root function from the linear function, we get:
\( y_1(x) = x - \sqrt{1 - x^2} \)
4Step 4: Combine the inverse trigonometric function
Now we will multiply our result, \(y_1(x)\), with the inverse trigonometric function \(\sin^{-1}x\). Be careful with the multiplication: only the square root function \(y_2(x)\) is multiplied by the inverse trigonometric function, not the whole expression. This results in the final function:
\( y(x) = x - \sqrt{1-x^2}\sin^{-1}x \)
5Step 5: Finalize the output
The given function consists of three major components - a linear function, a square root function, and an inverse trigonometric function. By carefully combining these components, we derived the final expression:
\( y(x) = x - \sqrt{1-x^2}\sin^{-1}x \)
Key Concepts
Differential CalculusSquare Root FunctionsLinear Functions
Differential Calculus
Differential calculus is a branch of mathematics that focuses on how things change and it is used to find the rate at which quantities change. The most fundamental operation in differential calculus is taking the derivative of a function, which measures the function's rate of change at any given point.
For example, if we have a function like the one from the exercise, \( y(x) = x - \sqrt{1-x^2}\sin^{-1}x \), to understand the rate at which \( y \) changes concerning \( x \), we would compute its derivative. This involves applying rules of differentiation such as the product rule, the chain rule, and the derivative of basic functions to find \( y'(x) \), the derivative of the function.
In the context of the exercise, understanding how to differentiate each component (linear, square root function, and inverse trigonometric function) is crucial. Each of these functions' derivatives is found differently, for instance, the derivative of a linear function is constant, while the derivative of a square root function involves the use of the chain rule and the derivative of an inverse trigonometric function often involves using implicit differentiation along with trigonometric identities.
For example, if we have a function like the one from the exercise, \( y(x) = x - \sqrt{1-x^2}\sin^{-1}x \), to understand the rate at which \( y \) changes concerning \( x \), we would compute its derivative. This involves applying rules of differentiation such as the product rule, the chain rule, and the derivative of basic functions to find \( y'(x) \), the derivative of the function.
In the context of the exercise, understanding how to differentiate each component (linear, square root function, and inverse trigonometric function) is crucial. Each of these functions' derivatives is found differently, for instance, the derivative of a linear function is constant, while the derivative of a square root function involves the use of the chain rule and the derivative of an inverse trigonometric function often involves using implicit differentiation along with trigonometric identities.
Square Root Functions
Square root functions, represented as \( f(x) = \sqrt{x} \), are a type of radical function where the independent variable \( x \) is under a square root.
These functions are defined for non-negative values of \( x \) because the square root of a negative number isn't defined in the set of real numbers. In the context of our exercise, which includes a square root function \( \sqrt{1-x^2} \), it's important to note that the domain is restricted to values of \( x \) for which the expression under the square root is non-negative, thus \( 1-x^2 \geq 0 \).
These functions are defined for non-negative values of \( x \) because the square root of a negative number isn't defined in the set of real numbers. In the context of our exercise, which includes a square root function \( \sqrt{1-x^2} \), it's important to note that the domain is restricted to values of \( x \) for which the expression under the square root is non-negative, thus \( 1-x^2 \geq 0 \).
Important Properties
- They typically represent half of a parabola that's been rotated 90 degrees.
- The graphs of square root functions always yield non-negative outputs (y-values).
- To find their derivatives, we use the chain rule in differential calculus, because they often don't just contain an \( x \) alone under the square root.
Linear Functions
Linear functions are perhaps the simplest form of functions we encounter in algebra. They are written in the form of \( f(x) = mx + b \), where \( m \) denotes the slope and \( b \) denotes the y-intercept.
The slope gives us the rate of change of the function, indicating how much the function's output changes for a given change in the input. The y-intercept tells us the value of \( y \) when \( x \) is zero, essentially where the graph of the function crosses the y-axis.
The slope gives us the rate of change of the function, indicating how much the function's output changes for a given change in the input. The y-intercept tells us the value of \( y \) when \( x \) is zero, essentially where the graph of the function crosses the y-axis.
Characteristics of Linear Functions
- They have a constant rate of change or slope.
- Their graph is always a straight line.
- They continue indefinitely in both directions, without any bends or curves.
- Their derivatives are constant because the rate of change is uniform.