5. continuity and differentiation — Maths STD 12 Science — Question

Differentiating w.r.t. $x$, we get 

$\frac{{dy}}{{dx}} = 2x{e^{mx}} + m{x^2}{e^{mx}}$

Again, $\frac{{{d^2}y}}{{d{x^2}}} = 2({e^{mx}} + mx{e^{mx}}) + m(2x{e^{mx}} + {x^2}m{e^{mx}})$

or $\frac{{{d^2}y}}{{d{x^2}}} = {e^{mx}}({m^2}{x^2} + 4mx + 2)$

Again, $\frac{{{d^3}y}}{{d{x^3}}} = {e^{mx}}[{m^3}{x^2} + 4{m^2}x + 2m + 2{m^2}x + 4m]$

$ = {e^{mx}}[{m^3}{x^2} + 6{m^2}x + 6m]$

$ \Rightarrow \frac{{{d^3}y}}{{d{x^3}}} = m{e^{mx}}({m^2}{x^2} + 6mx + 6)$.