Question

Find the derivative in the form dy/dx

Answer 1

`(dy)/(dx) = x{(3x)}^{ {x}^(2) } (ln(9 {x}^(2) ) + 1)`

Step-by-step explanation

The given equation is;

`y = {(3x)}^{ {x}^(2) }`

Take natural log of both sides:

`ln(y) = ln({(3x)}^{ {x}^(2) } )`

This implies that,

`ln(y) = {x}^(2) ln({(3x)})`

We differentiate to get,

`(1)/(y) (dy)/(dx) = ln(3x) (2x) + {x}^(2) ( (1)/(x) )`

This simplifies to

`(dy)/(dx) =y (2x ln(3x) + x)`We substitute for y to get:

`(dy)/(dx) = {(3x)}^{ {x}^(2) } (2x ln(3x) + x)`

Or

`(dy)/(dx) = x{(3x)}^{ {x}^(2) } (2ln(3x) + 1)`

`(dy)/(dx) = x{(3x)}^{ {x}^(2) } (ln(9 {x}^(2) ) + 1)`