Answer to Question #184475 in Calculus for Njabulo

Question

6. (Sections 1.3, 3.3, 7.6) Consider the R 2 − R function f defined by f(x, y) = ln xy and the R − R 2 function r defined by r(t) = ￾ t 2 , et
. Determine the value of (f ◦ r) 0 (1) by using the General Chain Rule (Theorem 7.6.1). Hints: • Determine (f ◦ r) 0 (1) directly from the formula given in Theorem 7.6.1. It is unnecessary to first determine (f ◦ r) 0 (t). • Check your answer by finding the composite function f ◦r, then finding the derivative (f ◦ r) 0 (t) and finally putting t = 1

Answer 1

Given

$f(x,y)=lnxy , r(t)-(t^2,e^t)$

$(foR)(t)=f(r(t))=f(t^2,e^t)=ln t^2e6t-lnt^2+lne^t=2lnt+t$

$\Rightarrow (foR)'(t)=(2lnt+t)'=\dfrac{2}{t}+1$

At t=1,

$\Rightarrow (foR)'(1)=2+1=3$