Question

How do I derive 3^(x^4)?

Answer 1

Hi andidious,

y=3^(x^4)
lny = x^4*ln3
using implicit differentiation,
(1/y)*(dy/dx)=4x^3*ln3
after simplifying it, you would get:
dy/dx = y*4x^3*ln3
= 3^(x^4)*(4x^3)*ln3

Hope it helps!

Answer 2

\[f(x) = 3^{x ^{4}}\]
logarithmic differentiation
\[\left( \ln f \right)\prime = \frac{ f \prime }{ f }\]
\[\ln f = \ln ( 3^{x ^{4}}) = x ^{4} . \ln 3 \]
\[\left( \ln f \right) \prime = ( x ^{4} . \ln 3 )\prime = 4.x ^{3}.\ln 3\]
\[f \prime = f.(\ln f)\prime = 3^{x ^{4}} . 4. x ^{3} .\ln3\]
almost the same with implicit differentiation !

Answer 3

thanks so much guys!