Menu Close

Please-i-need-your-help-if-y-tanx-tanx-tanx-find-dy-dx-at-4-




Question Number 5587 by sanusihammed last updated on 21/May/16
Please i need your help.    if   y  =  [tanx]^([tanx]^([tanx]) )    .   find  dy/dx  at  Π/4
$${Please}\:{i}\:{need}\:{your}\:{help}. \\ $$$$ \\ $$$${if}\:\:\:{y}\:\:=\:\:\left[{tanx}\right]^{\left[{tanx}\right]^{\left[{tanx}\right]} } \:\:\:.\:\:\:{find}\:\:{dy}/{dx}\:\:{at}\:\:\Pi/\mathrm{4} \\ $$
Commented by prakash jain last updated on 21/May/16
[tan x]=greatest integer (floor function)?  If not   y=(tan x)^((tan x)^((tan x)) )   ln y=(tan x)^((tan x)) ln tan x  ln ln y=tan xln (tan x)+ln (ln tan x)  differentiating using chain rule  (1/(ln y))∙(1/y)∙(dy/dx)=sec^2 xln (tan x)+tan x(1/(tan x))sec^2 x                                +(1/(ln (tan x)))∙(1/(tan x))∙sec^2 x  (dy/dx)=yln y(sec^2 x)(1+ln tan x+(1/(tan x∙ln tan x)))  (dy/dx)=(tan x)^((tan x)^((tan x)) ) ∙(tan x)^((tan x)) ln tan x(sec^2 x)(1+ln tan x+(1/(tan x∙ln tan x)))
$$\left[\mathrm{tan}\:{x}\right]={greatest}\:{integer}\:\left({floor}\:{function}\right)? \\ $$$$\mathrm{If}\:\mathrm{not}\: \\ $$$${y}=\left(\mathrm{tan}\:{x}\right)^{\left(\mathrm{tan}\:{x}\right)^{\left(\mathrm{tan}\:{x}\right)} } \\ $$$$\mathrm{ln}\:{y}=\left(\mathrm{tan}\:{x}\right)^{\left(\mathrm{tan}\:{x}\right)} \mathrm{ln}\:\mathrm{tan}\:{x} \\ $$$$\mathrm{ln}\:\mathrm{ln}\:{y}=\mathrm{tan}\:{x}\mathrm{ln}\:\left(\mathrm{tan}\:{x}\right)+\mathrm{ln}\:\left(\mathrm{ln}\:\mathrm{tan}\:{x}\right) \\ $$$${differentiating}\:{using}\:{chain}\:{rule} \\ $$$$\frac{\mathrm{1}}{\mathrm{ln}\:{y}}\centerdot\frac{\mathrm{1}}{{y}}\centerdot\frac{\mathrm{d}{y}}{\mathrm{d}{x}}=\mathrm{sec}^{\mathrm{2}} {x}\mathrm{ln}\:\left(\mathrm{tan}\:{x}\right)+\mathrm{tan}\:{x}\frac{\mathrm{1}}{\mathrm{tan}\:{x}}\mathrm{sec}^{\mathrm{2}} {x} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:+\frac{\mathrm{1}}{\mathrm{ln}\:\left(\mathrm{tan}\:{x}\right)}\centerdot\frac{\mathrm{1}}{\mathrm{tan}\:{x}}\centerdot\mathrm{sec}^{\mathrm{2}} {x} \\ $$$$\frac{{dy}}{{dx}}={y}\mathrm{ln}\:{y}\left(\mathrm{sec}^{\mathrm{2}} {x}\right)\left(\mathrm{1}+\mathrm{ln}\:\mathrm{tan}\:{x}+\frac{\mathrm{1}}{\mathrm{tan}\:{x}\centerdot\mathrm{ln}\:\mathrm{tan}\:{x}}\right) \\ $$$$\frac{{dy}}{{dx}}=\left(\mathrm{tan}\:{x}\right)^{\left(\mathrm{tan}\:{x}\right)^{\left(\mathrm{tan}\:{x}\right)} } \centerdot\left(\mathrm{tan}\:{x}\right)^{\left(\mathrm{tan}\:{x}\right)} \mathrm{ln}\:\mathrm{tan}\:{x}\left(\mathrm{sec}^{\mathrm{2}} {x}\right)\left(\mathrm{1}+\mathrm{ln}\:\mathrm{tan}\:{x}+\frac{\mathrm{1}}{\mathrm{tan}\:{x}\centerdot\mathrm{ln}\:\mathrm{tan}\:{x}}\right) \\ $$

Leave a Reply

Your email address will not be published. Required fields are marked *