Question

How do you find the derivative of `tanx/sinx`?

Answer 1

`d/dx[tan x/sinx] = d/dx[sec x] = sec x tan x`

Explanation:

Before we try to take a derivative, we could still simplify this expression.

Since `tan x = (sinx)/(cosx)`, we can write

`tanx/(sinx) = (cancel(sinx)/cosx * 1/(cancel(sinx)))/cancel((sinx/1 * 1/sinx)) = 1/cosx = sec x`

Also, we know by definition that `d/dx[sec x] = secx tanx`, thus giving us

`d/dx[tan x/sinx] = d/dx[sec x] = sec x tan x`