PARTIAL DIFFERENTIATION (Excercise)

\[\underline{PART\ -\ B}\]
\[1.\ If\ u\ =\ x^3\ +\ y^3\ ,\ find\ \frac{∂^2u}{∂x^2}\ \hspace{15cm}\]
\[\color {black}{Solution:}\ Given\ u\ =\ x^3\ +\ y^3\ \hspace{15cm}\]
\[\frac{∂}{∂x}\ (u)\ =\ \frac{∂}{∂x}( x^3)\ +\ \frac{∂}{∂x}(y^3)\ \ \hspace{10cm}\]
\[\frac{∂u}{∂x}\ =\ 3\ x^2\ +\ 0\ \frac{∂}{∂x}( y)\ \hspace{10cm}\]
\[Again\ Differentiate\ partially\ w.\ r.\ t.\ x\ on\ both\ sides\ \hspace{10cm}\]
\[\frac{∂}{∂x}(\frac{∂u}{∂x})\ =\ \frac{∂}{∂x}(3\ x^2)\ \hspace{10cm}\]
\[\frac{∂^2\ u}{∂x^2}\ =\ 3\ \frac{∂}{∂x}( x^2)\ \hspace{10cm}\]
\[\frac{∂^2\ u}{∂x^2}\ =\ 3\ (2\ x)\ \hspace{10cm}\]
\[\frac{∂^2\ u}{∂x^2}\ =\ 6\ x\ \hspace{10cm}\]