If \(y = \left|\sin\left(\dfrac{\pi}{4} - x\right)\right|\), then what is \(\dfrac{dy}{dx}\) at \(x = \dfrac{\pi}{4}\) equal to?
It does not exist
Let \(u = \sin\left(\dfrac{\pi}{4} - x\right)\); at \(x=\dfrac{\pi}{4}\), \(u=0\) and \(\dfrac{du}{dx} = -\cos\left(\dfrac{\pi}{4}-x\right) = -1\). Since \(u\) changes sign as \(x\) passes through \(\dfrac{\pi}{4}\) while \(\dfrac{du}{dx} \neq 0\) there, the left-hand derivative of \(y=|u|\) is \(-1\) and the right-hand derivative is \(+1\), which are unequal. Hence \(\dfrac{dy}{dx}\) does not exist at \(x=\dfrac{\pi}{4}\).
Similar Questions
The set of all points, where the function \({\rm{f}}\left( {\rm{x}} \right) = \sqrt {1 - {{\rm{e}}^{ - {{\rm{x}}^2}}}} \) is differentiable, is
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\( f(x)=\begin{cases} x^2\ln|x|, & x\neq 0,\\[4pt] 0, & x=0. \end{cases} \)
What is \(f'(0)\) equal to?
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Where k is an integer and [x] is the greatest integer function, isIf \({\rm{f}}\left( {\rm{x}} \right) = {\rm{x}}\left( {\sqrt {\rm{x}} - \sqrt {{\rm{x}} + 1} } \right)\) , then f(x) is
\({\rm{f}}\left( {\rm{x}} \right) = \left\{ {\begin{array}{*{20}{c}} {3{{\rm{x}}^2} + 12{\rm{x}} - 1,{\rm{\;\;}} - 1 \le {\rm{x}} \le 2}\\ {37 - {\rm{x}},{\rm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}}2 < {\rm{x}} \le 3} \end{array}} \right.\)
Which of the following statements is/are correct?
1. f(x) is increasing in the interval [-1, 2]
2. f(x) is decreasing in the interval (2, 3).
Select the correct answer using the code given below:
\({\rm{f}}\left( {\rm{x}} \right) = \left\{ {\begin{array}{*{20}{c}} {3{{\rm{x}}^2} + 12{\rm{x}} - 1,{\rm{\;\;}} - 1 \le {\rm{x}} \le 2}\\ {37 - {\rm{x}},{\rm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}}2 < {\rm{x}} \le 3} \end{array}} \right.\)
Which of the following statements are correct?
1. f(x) is continuous at x = 2
2. f(x) attains greatest value at x = 2
3. f(x) is differentiable at x = 2
Select the correct answer using the code given below:
Consider the following statements:
1. The function f(x) is continuous at x = 0
2. The function f(x) is continuous at \({\rm{x}} = \frac{{\rm{\pi }}}{2}\)
Which of the above statements is/are correct?Consider the following statements:
1. The function f(x) is differentiable at x = 0
2. The function f(x) is differentiable at \({\rm{x}} = \frac{{\rm{\pi }}}{2}\) .
Which of the above statements is/are correct?What is f’(4) equal to?
What is f’’(2.5) equal to?
Important Questions from Differentiability
What is the value of f'(x) at x = 4 from the following table of values?
x 1 2 3 4 f(x) 20 22 27 35 The set of all points, where the function \({\rm{f}}\left( {\rm{x}} \right) = \sqrt {1 - {{\rm{e}}^{ - {{\rm{x}}^2}}}} \) is differentiable, is
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