Derivatives: Find the derivative of x^(3^x)

[johnboy]

I'm really confused on this derivative..

. . .Find the derivative of x^(3^x)

Can I just do this?

. . .y = x^(3^x)

. . .ln(y) = ln(x^(3^x))

. . .ln(y) = (3^x)ln(x)

. . .1/y dy/dx = (3^x)ln(3)ln(x) + 3^x/x

. . .dy/dx = (x^(3^x)) ((3^x)ln(3)ln(x) + 3^x/x)

 

[johnboy]

can i just set it to y like i did there? can i just set the equation = to y if the directions just have x^3^x ? cause i know how to do logarithmic differentiation, i just didnt know if i could randomly throw in a y like that or not...

 

[pka]

can i just set the equation = to y if the directions just have x^3^x ?

Indeed it is! Your answer is correct.

 

[johnboy]

so, im allowed to set x^3^x equal to y then? thanks.

 

[soroban]

Hello, johnboy!

can i just do this?

\(\displaystyle y \:=\:x^{3^x}\)

\(\displaystyle \ln y \:=\:ln\left(x^{3^x}\right)\)

\(\displaystyle \ln y \:=\:3^x\cdot\ln x\)

\(\displaystyle \L\frac{1}{y}\left(\frac{dy}{dx}\right) \:= \:3^x\cdot\ln3\cdot\ln x\,+\,\frac{3^x}{x}\)

\(\displaystyle \L\frac{dy}{dx} \:= \:x^{3^x}\,\left(3^x\cdot\ln3\cdot\ln x\,+\,\frac{3^x}{x}\right)\)

Yes . . . you certainly can!


Wanna show-off? . . . Factor out \(\displaystyle \L\,\frac{3^x}{x}\)

\(\displaystyle \L\frac{dy}{dx} \:= \:\frac{x^{3^x}\cdot 3^x}{x}\left(x\cdot\ln3\cdot\ln x\,+\,1\right) \:= \:x^{^{(3^x-1)}}3^x(x\cdot\ln3\cdot\ln x\,+\,1)\)