Name: Differentiation of Exponential Functions
Uploaded: 2019-07-19
Duration: 12 min 3 s
Description: To know how to find the derivative of exponential functions or multiple functions containing the exponential term. Let's watch this video and explore!

Question

Prove that the function for $f (x) = \{\begin{cases} \frac{x}{|x|}, if x \neq 0 \\ 0, if x = 0 . \end{cases}$ is not differentiable at $x = 0$ .

Accepted Answer

The given function is $f (x) = \{\begin{cases} \frac{x}{|x|}, if x \neq 0 \\ 0, if x = 0 . \end{cases}$ which can be written as $f (x) = \{\begin{matrix} \frac{x}{x} = 1; if x > 0 \\ \frac{x}{- x} = - 1; if x < 0 \\ 0; x = 0 \end{matrix} \begin{array}{l} \end{array}$

[The absolute value or modulus of a real number $x$ ,denoted by $|x|$ ,is the non-negative value of $x$ without regard to its sign. Namely $|x| = x$ if $x$ is positive, $|x| = - x$ if $x$ is negative and $|0| = 0$ ]

Continuity at $x = 0$ ;

$LHL = \lim_{x \to 0^{-}} f (x) = \lim_{x \to 0} (- 1) = - 1$

$RHL = \lim_{x \to 0^{+}} f (x) = \lim_{x \to 0} 1 = 1$

Here, we get

$LHL \neq RHL$

Thus, function $f$ is discontinuous at $x = 0$ .

$∴ f$ is not differentiable at $x = 0$ .[ $∵$ Every differentiable function is continuous. So if a function is discontinuous at a point then it can not be differentiable at that point.]

Prove that the function for $f (x) = \{\begin{cases} \frac{x}{|x|}, if x \neq 0 \\ 0, if x = 0 . \end{cases}$ is not differentiable at $x = 0$ .

Important Questions on Continuity and Differentiability

Learn and Prepare for any exam you want !

Prove that the function for fx=xx , if x≠0 0 , if x=0.is not differentiable at x=0.

Important Questions on Continuity and Differentiability

Learn and Prepare for any exam you want !

Prove that the function for $f (x) = \{\begin{cases} \frac{x}{|x|}, if x \neq 0 \\ 0, if x = 0 . \end{cases}$ is not differentiable at $x = 0$ .