Question

Consider the R²−R function f defined by f(x,y)= lnxy and the R−R² function r defined by r(t) = (t²,e^t). Determine the value of (f◦r)′(1) by using the General Chain Rule

Answer 1

$f(x,y)=\ln xy$

$r(t)=(t^2,e^t)$

$(f\space o\space r)=f(r(t))=\ln t^2e^t$

$(f\space o \space r)'=\dfrac{1}{t^2 e^t}\times(2te^t+e^tt^2)=\dfrac{2}{t}+1$

$(f\space o \space r)'(1)=2+1=3$