Question

What will be the differentiation of this? d/dx (6tan(3x)sec ^2 (3x))

Answer 1

`(dy)/(dx)=18sec^2 3x(2tan^2 3x+sec^2 3x)`

Explanation:

Let ,

`y= 6tan(3x)sec ^2 (3x)`

We take,

`color(red)(y=u*v ,where, u=6tan3x and v=sec^2 3x`

Using chain rule:

`=>(du)/(dx)=6sec^2 3xd/(dx)(3x)=6sec^2 3x*3=18sec^2 3x`

`and (dv)/(dx)=2sec 3xd/(dx)(sec3x)=2sec3x*sec3xtan3x*3`

`=>(dv)/(dx)=6sec^2 3xtan3x`

Diff.w.r.t. `x` using product rule:

`color(blue)(d/(dx)(u*v)=u*(dv)/(dx)+v*(du)/(dx)`

`:.(dy)/(dx)=6tan3x(6sec^2 3xtan3x)+sec^2 3x(18sec^2 3x)`

`(dy)/(dx)=36sec^2 3xtan^2 3x+18sec^4 3x`

`(dy)/(dx)=18sec^2 3x(2tan^2 3x+sec^2 3x)`