Question and Answers Forum

All Questions      Topic List

Differentiation Questions

Previous in All Question      Next in All Question      

Previous in Differentiation      Next in Differentiation      

Question Number 11581 by Nayon last updated on 28/Mar/17

$$cAnyouprovethechAinrole??$$

Answered by linkelly0615 last updated on 28/Mar/17

$$yes\sim$$

Commented by linkelly0615 last updated on 28/Mar/17

$$\left(BELOW\right)$$

Answered by linkelly0615 last updated on 28/Mar/17

$$$$$$setthat{\left[f\left(g\left(x\right)\right)\right]}^{\prime }exist$$$$$$$${\left[f\left(g\left(x\right)\right)\right]}^{\prime }=\underset{h\to 0}{\lim }\frac{\left[f\left(g(x+h)\right)\right]-\left[f\left(g\left(x\right)\right)\right]}{h}$$$$\because g(x+h)=g\left(x\right)+{hg}^{\prime }\left(x\right)+\frac{h^2}{2}{g}^{″}\left(x\right)+......$$$$\therefore when‘‘h^{″}issmallenough$$$$then$$$$g(x+h)\approx g\left(x\right)+{hg}^{\prime }\left(x\right)$$$$\Rightarrow \left[f\left(g(x+h)\right)\right]\approx \left[f(g\left(x\right)+{hg}^{\prime }\left(x\right))\right]$$$$if‘‘{hg}^{\prime }{\left(x\right)}^{″}issmallenough$$$$then$$$$\left[f(g\left(x\right)+{hg}^{\prime }\left(x\right))\right]\approx [f\left(g\left(x\right)\right)+\left({hg}^{\prime }\left(x\right)\right)f^{\prime }\left(g\left(x\right)\right)]$$$$...$$$$\Rightarrow$$$${\left[f\left(g\left(x\right)\right)\right]}^{\prime }=\underset{h\to 0}{\lim }\frac{\left[f\left(g(x+h)\right)\right]-\left[f\left(g\left(x\right)\right)\right]}{h}$$$$=\underset{h\to 0}{\lim }\frac{\left[f(g\left(x\right)+{hg}^{\prime }\left(x\right))\right]-\left[f\left(g\left(x\right)\right)\right]}{h}$$$$=\underset{h\to 0}{\lim }\frac{[f\left(g\left(x\right)\right)+\left({hg}^{\prime }\left(x\right)\right)f^{\prime }\left(g\left(x\right)\right)]-\left[f\left(g\left(x\right)\right)\right]}{h}$$$$=f^{\prime }\left(g\left(x\right)\right)g^{\prime }\left(x\right)$$

Commented by linkelly0615 last updated on 28/Mar/17

$$Ithinkitmightbewhatyouwant.$$

Commented by Nayon last updated on 29/Mar/17

$$Canyouproveitanyeasyway$$$$withoutusinganyseries?$$

Commented by linkelly0615 last updated on 29/Mar/17

Commented by linkelly0615 last updated on 29/Mar/17

$$Maybeitisokay$$$$ifnot...Iwilltry...$$$$\mathrm{actually},Iamtryingtofind$$$$somediffrentwaystoprove$$$$thatrule...$$$$\left(Includingacrazyway\right)$$$$$$

Commented by Nayon last updated on 29/Mar/17

$$itisnotseenproperlly....please$$$$commentaclearone..$$

Commented by linkelly0615 last updated on 29/Mar/17

$$itcanzoominbyyourfinger...$$$$$$

Commented by linkelly0615 last updated on 29/Mar/17

Terms of Service

Privacy Policy

Contact: info@tinkutara.com