Differentiation Question 386
Question: If $ u=x^{2}+y^{2} $ and $ x=s+3t, $
$ y=2s-t, $ then $ \frac{d^{2}u}{ds^{2}}= $
[Orissa JEE 2002]
Options:
A) 12
B) 32
C) 36
D) 10
Show Answer
Answer:
Correct Answer: D
Solution:
$ u=x^{2}+y^{2}, $
$ x=s+3t, $
$ y=2s-t $
Now $ \frac{dx}{ds}=1, $
$ \frac{dy}{ds}=2 $
…..(i) $ \frac{d^{2}x}{ds^{2}}=0, $
$ \frac{d^{2}y}{ds^{2}}=0 $
……(ii) Now $ u=x^{2}+y^{2} $ , $ \frac{du}{ds}=2x.\frac{dx}{ds}+2y.\frac{dy}{ds} $
$ \frac{d^{2}u}{ds^{2}}=2{{( \frac{dx}{ds} )}^{2}}+2x\frac{d^{2}x}{ds^{2}}+2{{( \frac{dy}{ds} )}^{2}}+2y( \frac{d^{2}y}{ds^{2}} ) $
From (i) and (ii), $ \frac{d^{2}u}{ds^{2}}=2\times 1+0+2\times 4+0=10 $ .
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