Question

How to find the derivative of f(x)=(sinx)^x ?

Answer 1

\[f(x)=(sinx)^x\]

Answer 2

Use the quotient rule:

f (x) = sin (x) / x

dy/dx = [ d/dx (sin (x) ) * x - (sin (x) ) * d/dx (x) ] / x²

dy/dx = [ cos (x) * x - (sin (x) ) ] / x²

dy/dx = [ xcos (x) - sin (x) ] / x²

dy/dx = ( x cos (x) / x² ) - ( sin (x) / x² )

dy/dx = ( cos (x) / x ) - ( sin (x) / x² )

Answer 3

First:
\[\LARGE lny=ln((sinx)^x)\]
\[\LARGE lny=xln(sinx)\]
then take the derivative and use product and chain rule ;)

\[\LARGE \frac{dy}{dx} \frac{1}{y}=x(\frac{1}{sinx})cosx + ln(sinx)\]
then simplfy

\[\LARGE \frac{dy}{dx}\frac{1}{y} = xcotx+lnsinx\]
\[\LARGE \frac{dy}{dx}=y[xcotx+lnsinx]\]

plug in for y

\[\LARGE \frac{dy}{dx}= (sinx)^x[xcotx+lnsinx]\]

Answer 4

welcome 2 os

Answer 5

Thanks guys for the help @Dakota005 @Luigi0210 :)

Answer 6

np do u have anymore questions?

Answer 7

yes i have but i am still doing it @Dakota005 like this one \[y=(4x-7)^3(3-5x)^7\]

Answer 8

that 1 i dont undestand. i think @Luigi0210 is smart nonlike me. @Luigi0210 can u please help @boyb39

Answer 9

Oh my, that looks troublesome