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Chapter 5: Problem 52

If \(y=\sin ^{-1}\left(x \sqrt{1-x}-\sqrt{x-x^{3}}\right)\), find \(\frac{d y}{d x}\)

Short Answer

Expert verified
\(\frac{dy}{dx} = \frac {1}{\sqrt{1 - (f(x) - g(x))^2}} \cdot (\frac{df}{dx} - \frac{dg}{dx})\). Replace the functions \(f(x), g(x)\) and their derivatives with the corresponding expressions, and simplify to get the final answer.

Step by step solution

01

Identify and comprehend the individual functional components

Here, we identify two functions, \(f(x) = x \sqrt{1-x}\) and \(g(x) = \sqrt{x-x^{3}}\) and we combine them in an inverse sine function. So, the overall function becomes \(y=\sin ^{-1}(f(x)-g(x))\).
02

Differentiate the individual components

Using the product rule on \(f(x)\), \(\frac{df}{dx} = \sqrt{1-x} + x \cdot \frac{1}{2 \sqrt{1-x}} \cdot (-1)\). Differentiating \(g(x)\), \(\frac{dg}{dx} = \frac{1}{2\sqrt{x-x^3}} \cdot (1-3x^2)\).
03

Differentiate the Inverse Sine function

The derivative of \(\sin ^{-1}(x)\) using the chain rule is \(\frac{1}{\sqrt{1-x^2}}\) but since we are dealing with \(\sin ^{-1}(f(x) - g(x))\) instead, we have to take into account the derivate of the entirety of \((f(x)-g(x))\) - which is \(\frac{df}{dx} - \frac{dg}{dx}\). So, \(\frac{dy}{dx} = \frac {1}{\sqrt{1 - (f(x) - g(x))^2}} \cdot (\frac{df}{dx} - \frac{dg}{dx})\).
04

Insert the derived individual components in the elementary function

By substituting for \(f(x)\), \(g(x)\), \(\frac{df}{dx}\), and \(\frac{dg}{dx}\) into above expression, we get the derivative of \(y\) with respect to \(x\).

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Key Concepts

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

Chain Rule
When working with derivatives, the chain rule is vital for handling composite functions. A composite function is where one function is inside another, such as our inverse sine function. The chain rule tells us how to differentiate these. If you have a function like \( y = \sin^{-1}(u(x)) \), the derivative is \( \frac{d}{dx}(\sin^{-1}(u)) = \frac{1}{\sqrt{1-u^2}} \cdot \frac{du}{dx} \).
This means we first find the derivative of the inside function \( u(x) \), and then multiply it by the derivative of the inverse sine function.
Understanding the chain rule helps us tackle more complex derivatives efficiently by breaking them into simpler parts.
Product Rule
The product rule is crucial when differentiating products of two functions. If you have two functions multiplied together, say \( u(x) \) and \( v(x) \), then the derivative \( \frac{d}{dx}(u \cdot v) \) is \( u'v + uv' \).
In our problem, the function \( f(x) = x \sqrt{1-x} \) is the product of \( x \) and \( \sqrt{1-x} \).
Using the product rule, we find \( \frac{df}{dx} = \sqrt{1-x} + x \cdot \frac{1}{2\sqrt{1-x}} \cdot (-1) \).
This approach allows us to break down complex derivatives into manageable parts by focusing on each function separately.
Inverse Sine
The inverse sine function, represented as \( \sin^{-1}(x) \), is one type of inverse trigonometric function. Its derivative is an essential formula: \( \frac{d}{dx}(\sin^{-1}(x)) = \frac{1}{\sqrt{1-x^2}} \).
In our exercise, the inverse sine takes a more complex form, \( \sin^{-1}(f(x) - g(x)) \).
To find the derivative, we apply the chain rule: first calculating the derivative of the inside portion \( f(x) - g(x) \), and then applying the inverse sine derivative formula.
This step-by-step differentiation helps students understand how inverse functions behave and how to handle them easily.
Function Composition
Function composition involves combining two or more functions, where the output of one serves as the input for another.
In our example, the composition is seen in \( \sin^{-1}(f(x) - g(x)) \).
First, we identify how each function contributes. Here, \( f(x) = x \sqrt{1-x} \) and \( g(x) = \sqrt{x-x^3} \) are subtracted, and then wrapped into the inverse sine.
This helps to structure the problem, making it less daunting by breaking it down into digestible parts.
Understanding function composition is key to mastering more complicated calculus problems, as it helps to untangle intertwined functions.

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Most popular questions from this chapter

If \(x=\sin ^{-1}\left(\frac{3 \sin t+4 \cos t}{5}\right)\) and \(y=\sin ^{-1}\left(\frac{6 \cos t+8 \sin t}{10}\right)\), find \(\frac{d y}{d x}\).

Let \(f\) be a function for which \(f^{\prime}(x)=x^{2}+1\) If \(y=f\left(\sin \left(x^{3}\right)\right)\), find \(\frac{d y}{d x}\).

If \(x=a\left(t+\frac{1}{t}\right)\) and \(y=a\left(t-\frac{1}{t}\right)\), then prove that \(\frac{d y}{d x}=\frac{x}{y}\).

If \(y=x+\tan x\), prove that \(\cos ^{2} x \frac{d^{2} y}{d x^{2}}-2 y+2 x=0\)

If \(y=\log \left(x+\sqrt{x^{2}+1}\right)\), find \(\frac{d y}{d x}\)
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