Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 15 - Differentiation in Several Variables - 15.3 Partial Derivatives - Exercises - Page 782: 68


$$ u_{xx}= -\frac{1}{2} t^{-3/2}+\frac{1}{4}x^2t^{-5/2}e^{-x^2/4t}.$$

Work Step by Step

Since $ u(x,t)=t^{-1/2}e^{-x^2/4t}$, by using the chain rule, we get $$ u_x=t^{-1/2}e^{-x^2/4t}(-2x/4t)=-\frac{1}{2}xt^{-3/2}e^{-x^2/4t}.$$ Deriving again, we get: $$ u_{xx}=-\frac{1}{2} t^{-3/2}e^{-x^2/4t}-\frac{1}{2}xt^{-3/2}e^{-x^2/4t}(-2x/4t)\\=-\frac{1}{2} t^{-3/2}+\frac{1}{4}x^2t^{-5/2}e^{-x^2/4t}.$$
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