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Chapter 2: Problem 68

Find the derivatives from the left and from the right at \(x=1\) (if they exist). Is the function differentiable at \(x=1 ?\) \(f(x)=\sqrt{1-x^{2}}\)

Short Answer

Expert verified
The derivatives from the left and the right at \(x=1\) are both undefined, hence the function \(f(x)=\sqrt{1-x^{2}}\) is not differentiable at \(x=1\).

Step by step solution

01

Title: Calculation of the Derivative

Start by finding the derivative of the function. For this function, the power rule of a function can be used. The derivative of \(f(x)=\sqrt{1-x^{2}}\) can be calculated as \(f'(x) = -\frac{x}{\sqrt{1-x^{2}}}\)
02

Title: Calculation of Derivatives from the Left and Right

After finding the derivative, the next step is to calculate the derivatives from the left and the right at \(x=1\). This can be obtained by finding the limits. For the left-hand derivative, \lim_{h \to 0}\frac{f(1-h) - f(1)}{h} needs to be calculated. Given \(f(1)= \sqrt{1-1^{2}} = 0\), the left-hand derivative becomes undefined. Similarly, for the right-hand derivative, \lim_{h \to 0}\frac{f(1+h) - f(1)}{h} also results in undefined as it also involves division by zero.
03

Title: Determine if the Function is Differentiable

The function is said to be differentiable at a point if the left derivative and the right derivative at that point are equal. In the previous step, it was found that the left and the right derivatives at \(x=1\) are both undefined. Therefore, the function is not differentiable at \(x=1\).

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