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Chapter 4: Problem 208

$$ \text { Differentiate } \sin ^{-1} \frac{1-x}{1+x} \text { w.r.t. } \sqrt{x} $$

Short Answer

Expert verified
\( \frac{4x\sqrt{x}}{(1+x)^2 \sqrt{1 - (\frac{1-x}{1+x})^2}} \)

Step by step solution

01

Differentiate the Outer Function

Differentiate the inverse sine function using its derivative: \( \frac{d}{dx} \sin^{-1}(u) = \frac{1}{\sqrt{1 - u^2}} \cdot u' \). So, the derivative of \( \sin ^{-1} \frac{1-x}{1+x} \) with respect to \( \frac{1-x}{1+x} \) is \( \frac{1}{\sqrt{1 - (\frac{1-x}{1+x})^2}} \).
02

Differentiate the Inner Function

Now find the derivative of the inner function \( \frac{1-x}{1+x} \) with respect to \( x \). We would use the quotient rule which states that \( (f/g)' = f'g - fg' / g^2 \). Applying this gives \( \frac{(1+x) - (1-x)}{(1+x)^2} = \frac{2x}{(1+x)^2} \).
03

Differentiate the Outer Function with respect to \( \sqrt{x} \)

Now, we differentiate \( x \) with respect to \( \sqrt{x} \). We can express \( x \) as \( \sqrt{x}^2 \), and by applying chain rule, we get the derivative as \( 2 \cdot \sqrt{x} \) .
04

Piece it all together

Multiply all the derivative parts together: \( \frac{1}{\sqrt{1 - (\frac{1-x}{1+x})^2}} \cdot \frac{2x}{(1+x)^2} \cdot 2\sqrt{x} \). Simplifying the expression, we get: \( \frac{4x\sqrt{x}}{(1+x)^2 \sqrt{1 - (\frac{1-x}{1+x})^2}} \). This is the derivative of the given function with respect to \( \sqrt{x} \).

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Most popular questions from this chapter

$$ y=\sqrt{x \sin x \sqrt{1-e^{x}}} $$

Draw the graph of the function \(y=|x-1|+|x-2|\) in the interval \([0,3]\) and discuss the continuity and differentiability of the function in this interval. \\{Ans. continuous, not differentiable at \(x=1 \& 2\\}\)

$$ y=0.4\left(\cos \frac{2 x+1}{2}-\sin 0.8 x\right)^{2} $$

$$ \begin{aligned} &\text { If } \begin{aligned} f(x) &=a x^{2}+b, \quad x \leq 1 \\ &=b x^{2}+a x+c, \quad x>1, \end{aligned}\\\ &\text { where } b \neq 0 \text { . Find } a \text { and } c \text { such that } f(x) \text { is continuous and differentiable at } x=1 \text { . } \end{aligned} $$
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