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

Differentiate

|\cos x|

with respect to

x

, Is the function differentiable

?

What you can say about the differentiability of

\cos |x|

.

Accepted Answer

The given function is $|\cos x|$ .

Let $f (x) = |\cos x|$

On differentiating both sides with respect to $x$ , we get

$\frac{d}{d x} f (x) = \frac{d}{d x} |\cos x|$

$f' (x) = \frac{\cos x}{|\cos x|} \cdot \frac{d}{d x} \cos x [∵ \frac{d}{d x} |x| = \frac{x}{|x|}; x \neq 0]$

$f' (x) = \frac{\cos x}{|\cos x|} \cdot (- \sin x) [∵ \frac{d}{d x} \cos x = - \sin x]$

$f' (x) = \frac{- \sin x \cos x}{|\cos x|}$ , provided $\cos x \neq 0$

$f' (x) = \frac{- \sin x \cos x}{|\cos x|}$ , provided $x \in R$ - {odd multiple of $\frac{π}{2}$ } or $x \neq (2 n + 1) \frac{π}{2} \forall n \in I$ .

Since $f' (x)$ does not exists at $x = (2 n + 1) \frac{π}{2}, \forall n \in I$ .

But $x = (2 n + 1) \frac{π}{2}, n \in I \in$ domain of $\cos x$ , which is all real $R$ .

Hence $|\cos x|$ is not differentiable.

Also $\cos |x| = \{\begin{cases} \cos x; x \geq 0 \\ \cos (- x); x < 0 \end{cases} = \{\begin{cases} \cos x; x \geq 0 \\ \cos x; x < 0 \end{cases}$

The cosine function is continuous everywhere. if $f (x)$ and $g (x)$ are continuous at some points $p$ , $f (g (x))$ is also continuous and differentiable at that point.

$∴ \cos |x| = \cos x,$ which is differentiable everywhere.

Hence $\cos |x|$ is differentiable everywhere.

Differentiate $|\cos x|$ with respect to $x$ , Is the function differentiable $?$ What you can say about the differentiability of $\cos |x|$ .

Important Questions on Continuity and Differentiability

Learn and Prepare for any exam you want !

Differentiate cos x with respect to x, Is the function differentiable? What you can say about the differentiability of cosx.

Important Questions on Continuity and Differentiability

Learn and Prepare for any exam you want !

Differentiate $|\cos x|$ with respect to $x$ , Is the function differentiable $?$ What you can say about the differentiability of $\cos |x|$ .