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Differentiation Marathon!
Posted: Sun Apr 04, 2021 11:29 pm
by Anindya Biswas
Thanks for the idea from Asif Hossain. Marathon rules are same. Don't let a problem stay unsolved for more than $2$ days. And post a new derivative problem as soon as one gets solved.
Let's start the journey by a simple one.
$\text{Problem 0 :}$
Find the derivative of the function \[e^{e^{e^{e^x}}}\]
Re: Differentiation Marathon!
Posted: Mon Apr 05, 2021 12:44 pm
by Asif Hossain
Applying chain rule $f'(x)=e^{e^{e^{e^x}}}{e^{e^{e^x}}}{e^{e^x}}e^x$
Problem 1
Posted: Mon Apr 05, 2021 2:48 pm
by Asif Hossain
Find $f''(x)$ where $f(x)=e^{e^{e^{e^x}}}$
Solution to Problem 1
Posted: Mon Apr 05, 2021 5:03 pm
by Anindya Biswas
Asif Hossain wrote: ↑Mon Apr 05, 2021 2:48 pm
Find $f''(x)$ where $f(x)=e^{e^{e^{e^x}}}$
Problem 2
Posted: Mon Apr 05, 2021 5:45 pm
by Anindya Biswas
Prove that \[\frac{d}{dx}\sin{x}=\cos{x}\]
Re: Differentiation Marathon!
Posted: Mon Apr 05, 2021 9:41 pm
by Mehrab4226
A formula we are going to use,
$\sin A - \sin B = 2\sin \frac{A-B}{2} \cos \frac{A+B}{2}$
Now,
$\frac{d}{dx}\sin X = \lim _{h \to 0} \frac{\sin (x+h)-\sin x}{h}$
$= \lim _{h \to 0} \frac{2\sin \frac{x+h-x}{2} \cos \frac{2x+h}{2}}{h}$
$= \lim _{h \to 0} \frac{2\sin \frac{h}{2} \cos \frac{2x+h}{2}}{h}$
$= \lim _{h \to 0} \frac{2\sin \frac{h}{2} \cos \frac{2x+h}{2}}{2\frac{h}{2}}$
$= \lim _{\frac{h}{2} \to 0} \frac{2\sin \frac{h}{2}}{2\frac{h}{2}}\times \lim _{h \to 0}\cos (x+\frac{h}{2})$
$=\lim _{\frac{h}{2} \to 0} \frac{\sin \frac{h}{2}}{\frac{h}{2}} \times \lim _{h \to 0}\cos (x+\frac{h}{2})$
$=1 \times \cos (x+0) $ [since $\lim _{x \to 0} \frac{sin x}{x}=1$]
$=\cos x$
Problem 3
Posted: Mon Apr 05, 2021 11:08 pm
by Asif Hossain
Here goes another easy one
Differentiate $f(x)=x^{(sin(cos(x))}$
Solution to Problem 3
Posted: Tue Apr 06, 2021 12:14 am
by Anindya Biswas
Asif Hossain wrote: ↑Mon Apr 05, 2021 11:08 pm
Here goes another easy one
Differentiate $f(x)=x^{(sin(cos(x))}$
Problem 4
Posted: Tue Apr 06, 2021 1:08 am
by Anindya Biswas
Let $f$ be an infinitely differentiable function. Prove that, \[\frac{d^n}{dx^n}\left(e^xf(x)\right)=e^x\sum_{k=0}^{n}{n\choose k}f^{(k)}(x)\] for all nonnegative integer $n$.
Re: Problem 4
Posted: Tue Apr 06, 2021 9:37 am
by Mehrab4226
Anindya Biswas wrote: ↑Tue Apr 06, 2021 1:08 am
Let $f$ be an infinitely differentiable function. Prove that, \[\frac{d^n}{dx^n}\left(e^xf(x)\right)=e^x\sum_{k=0}^{n}{n\choose k}f^{(k)}(x)\] for all nonnegative integer $n$.
For the right hand side,
If k=2, does $f^{(k)}(x)$ mean
$f^{(2)}(x)=f''(x)$ this? or,
$f^{(2)}(x)=f^{2}(x)=f(f(x))$ this?