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Question Number 184184 by Mastermind last updated on 03/Jan/23

y=(sinx)^x Differentiate

$$\mathrm{y}=\left(\mathrm{sinx}\right)^{\mathrm{x}} \\ $$$$ \\ $$$$\mathrm{Differentiate} \\ $$

Answered by SEKRET last updated on 03/Jan/23

y = (sin(x))^x ln(y) = x ∙ ln(sin(x)) ((y′)/y) = ln(sin(x)) + x∙ ((cos(x))/(sin(x))) y ′ = (sin(x))^x ∙(ln(sin(x))+ x∙ctg(x))

$$\:\:\:\:\:\boldsymbol{\mathrm{y}}\:=\:\left(\boldsymbol{\mathrm{sin}}\left(\boldsymbol{\mathrm{x}}\right)\right)^{\boldsymbol{\mathrm{x}}} \\ $$$$\:\:\:\:\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{y}}\right)\:=\:\boldsymbol{\mathrm{x}}\:\centerdot\:\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{sin}}\left(\boldsymbol{\mathrm{x}}\right)\right) \\ $$$$\:\:\:\:\:\:\:\frac{\boldsymbol{\mathrm{y}}'}{\boldsymbol{\mathrm{y}}}\:=\:\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{sin}}\left(\boldsymbol{\mathrm{x}}\right)\right)\:+\:\boldsymbol{\mathrm{x}}\centerdot\:\frac{\boldsymbol{\mathrm{cos}}\left(\boldsymbol{\mathrm{x}}\right)}{\boldsymbol{\mathrm{sin}}\left(\boldsymbol{\mathrm{x}}\right)} \\ $$$$\:\:\:\:\boldsymbol{\mathrm{y}}\:'\:=\:\left(\boldsymbol{\mathrm{sin}}\left(\boldsymbol{\mathrm{x}}\right)\right)^{\boldsymbol{\mathrm{x}}} \centerdot\left(\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{sin}}\left(\boldsymbol{\mathrm{x}}\right)\right)+\:\boldsymbol{\mathrm{x}}\centerdot\boldsymbol{\mathrm{ctg}}\left(\boldsymbol{\mathrm{x}}\right)\right) \\ $$

Commented by Mastermind last updated on 03/Jan/23

V. good of you, everything is correct thank you so much for answering it

$$\mathrm{V}.\:\mathrm{good}\:\mathrm{of}\:\mathrm{you},\:\mathrm{everything}\:\mathrm{is}\:\mathrm{correct} \\ $$$$\mathrm{thank}\:\mathrm{you}\:\mathrm{so}\:\mathrm{much}\:\mathrm{for}\:\mathrm{answering}\:\mathrm{it} \\ $$

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