Question

6. e sin 5x

Answer 1

The given function is e x sin5x.

Let y= e x sin5x.

Differentiate function y with respect to x.

dy dx = d( e x sin5x ) dx dy dx = d( e x ) dx .sin5x+ d( sin5x ) dx . e x dy dx = e x .sin5x+cos5x d( 5x ) dx . e x dy dx = e x .sin5x+5. e x .cos5x

Again, differentiate with respect to x.

d dx ( dy dx )= d( e x .sin5x+5. e x .cos5x ) dx d 2 y d x 2 = d( e x .sin5x ) dx + d( 5. e x .cos5x ) dx d 2 y d x 2 = e x sin5x+cos5x.5. e x +5( e x cos5x−sin5x.5. e x ) d 2 y d x 2 = e x sin5x+5 e x cos5x+5 e x cos5x−25 e x sin5x

Simplify further,

d 2 y d x 2 =10 e x cos5x−24 e x sin5x d 2 y d x 2 =2 e x ( 5cos5x−12sin5x )

Thus, the second order derivative of the given function is 2 e x ( 5cos5x−12sin5x ).