Question

Find the right-hand derivative of $f\left( x \right) = \left[ x \right]\sin \pi x$ at $x = n$, where $n \in I$.

Answer 1

Here in this question, we have to find the right hand derivative of given function.to solve this by using the formula of right hand derivative i.e., $RHD = \mathop {\lim }\limits_{x \to {c^ + }} \dfrac{{f(x) - f(c)}}{{x - c}}$, on substituting the function of x and c and on further simplification we get the required solution.

Complete step by step answer:
The idea of a limit is a basis of all calculus. The limit of a function is defined as let $f(x)$be a function defined on an interval that contains$x = a$. Then we say that$\mathop {\lim }\limits_{x \to a} f(x) = L$, if for every $\varepsilon > 0$there is some number $\delta > 0$such that $\left| {f(x) - L} \right| < \varepsilon$whenever$0 < \left| {x - a} \right| < \delta$.
Consider the given function:
$\Rightarrow \,\,f\left( x \right) = \left[ x \right]\sin \pi x$ ---------(1)
Now, we haver to find the right hand derivative.
i.e., $RHD = \mathop {\lim }\limits_{x \to {c^ + }} \dfrac{{f(x) - f(c)}}{{x - c}}$-------(2)
Here to examine the differentiability, take some substitution for $\mathop {\lim }\limits_{x \to {c^ + }} f(x)$ put $x = n + h$ and change the limit as $x \to {n^ + }$by $h \to 0$, then Equation (2) becomes
$\Rightarrow \,\,RHD = \mathop {\lim }\limits_{h \to 0} \dfrac{{f(n + h) - f(n)}}{h}$--------(3)
On substituting equation (2) in (3), then
$\Rightarrow \,\,RHD = \mathop {\lim }\limits_{h \to 0} \dfrac{{\left[ {n + h} \right]\sin \pi \left( {n + h} \right) - \left[ n \right]\sin \pi n}}{h}$
$\Rightarrow \,\,RHD = \mathop {\lim }\limits_{h \to 0} \dfrac{{\left( {n + h} \right)\sin \left( {n\pi + h\pi } \right) - n\sin \pi n}}{h}$
$\Rightarrow \,\,RHD = \mathop {\lim }\limits_{h \to 0} \dfrac{{\left( {n + h} \right)\sin \left( {n\pi + h\pi } \right) - n\sin n\pi }}{h}$
Now use the sine addition identity: $sin\left( {A + B} \right) = sinA{\text{ }}cosB + cosA\,sinB$ applying in the above equation, then
$\Rightarrow \,\,RHD = \mathop {\lim }\limits_{h \to 0} \dfrac{{\left( {n + h} \right)\left( {\sin \left( {n\pi } \right)\cos \left( {h\pi } \right) + \cos \left( {n\pi } \right)\sin \left( {h\pi } \right)} \right) - n\sin n\pi }}{h}$
As we know the value of $\cos n\pi = 1$ and $\sin \left( {n\pi } \right) = 0$. On substituting, we get
$\Rightarrow \,\,RHD = \mathop {\lim }\limits_{h \to 0} \dfrac{{\left( {n + h} \right)\left( {0 \cdot 1 + \cdot 1\sin \left( {h\pi } \right)} \right) - n.0}}{h}$
$\Rightarrow \,\,RHD = \mathop {\lim }\limits_{h \to 0} \dfrac{{\left( {n + h} \right)\left( {0 + \sin \left( {h\pi } \right)} \right) - 0}}{h}$
$\Rightarrow \,\,RHD = \mathop {\lim }\limits_{h \to 0} \dfrac{{(n + h)(\sin (h\pi ))}}{h}$
$\Rightarrow \,\,RHD = \mathop {\lim }\limits_{h \to 0} \dfrac{{n\sin (h\pi ) + h\sin (h\pi )}}{h}$
On simplification, we get
$\Rightarrow \,\,RHD = \mathop {\lim }\limits_{h \to 0} \left( {\dfrac{{n\sin (h\pi )}}{h} + \dfrac{{h\sin (h\pi )}}{h}} \right)$
$\Rightarrow \,\,RHD = \mathop {\lim }\limits_{h \to 0} \dfrac{n}{h}\sin (h\pi ) + \mathop {\lim }\limits_{h \to 0} \dfrac{h}{h}\sin (h\pi )$
$\Rightarrow \,\,RHD = \mathop {\lim }\limits_{h \to 0} \dfrac{n}{h}\sin (h\pi ) + \mathop {\lim }\limits_{h \to 0} \sin (h\pi )$
On applying limit, we get the first term in the form of $\dfrac{0}{0}$ form so we apply the L’Hospitals rule to the first term.
$\Rightarrow \,\,RHD = n\mathop {\lim }\limits_{h \to 0} h\,\cos (h\pi ) + \mathop {\lim }\limits_{h \to 0} \sin (h\pi )$
Now when we apply the limit to the above function
$\Rightarrow \,\,RHD = n.0\,\cos (0.\pi ) + \sin (0.\pi )$
Hence on simplification we get
$\Rightarrow \,\,RHD = 0$
Hence, the right hand derivative of the given function is 0.

Note: The problem is related to the limits we have to find the right hand limit. We must know the formula $RHD = \mathop {\lim }\limits_{x \to {c^ + }} \dfrac{{f(x) - f(c)}}{{x - c}}$ and we have to consider the given function and then we apply the limit to it. Since the given function is a trigonometry function we must know about the standard formulas of trigonometry.