Question

5. x3 log x

Answer 1

The given function is x 3 logx.

Let y= x 3 logx.

Differentiate function y with respect to x.

dy dx = d( x 3 logx ) dx dy dx = d( x 3 ) dx .logx+ d( logx ) dx . x 3 dy dx =3 x 2 logx+ 1 x . x 3 dy dx =3 x 2 logx+ x 2

Again differentiate with respect to x.

d dx ( dy dx )= d( 3 x 2 logx+ x 2 ) dx d 2 y d x 2 = d( 3 x 2 .logx ) dx + d( x 2 ) dx d 2 y d x 2 =3( 2x.logx+ 1 x x 2 )+2x d 2 y d x 2 =6xlogx+3x+2x

Simplify further,

d 2 y d x 2 =6xlogx+5x d 2 y d x 2 =x( 6logx+5 )

Thus, the second order derivative of the given function is x( 5+6logx ).