Differentiation Question 360
Question: If $ y=x^{2}e^{mx} $ , where m is a constant, then $ \frac{d^{3}y}{dx^{3}}= $
[MP PET 1987]
Options:
A) $ me^{mx}(m^{2}x^{2}+6mx+6) $
B) $ 2m^{3}xe^{mx} $
C) $ me^{mx}(m^{2}x^{2}+2mx+2) $
D) None of these
Show Answer
Answer:
Correct Answer: A
Solution:
$ y=x^{2}e^{mx} $
Differentiating w.r.t. $ x $ , we get $ \frac{dy}{dx}=2xe^{mx}+mx^{2}e^{mx} $
Again, $ \frac{d^{2}y}{dx^{2}}=2(e^{mx}+mxe^{mx})+m(2xe^{mx}+x^{2}me^{mx}) $
or $ \frac{d^{2}y}{dx^{2}}=e^{mx}(m^{2}x^{2}+4mx+2) $
Again, $ \frac{d^{3}y}{dx^{3}}=e^{mx}[m^{3}x^{2}+4m^{2}x+2m+2m^{2}x+4m] $
$ =e^{mx}[m^{3}x^{2}+6m^{2}x+6m] $
$ \Rightarrow \frac{d^{3}y}{dx^{3}}=me^{mx}(m^{2}x^{2}+6mx+6) $ .
BETA
