Question

How do you find the derivative of `y=14tanxcosx+10cscx`?

Answer 1

`dy/dx=14cosx-10csc x cot x`

Explanation:

In order to differentiate this function, we will need to use two different differentiation rules:

1. The Chain Rule
2. The Product Rule

Chain rule:

`[f(x)]^n=n[f(x)]^(n-1)xxf'(x)`

Product rule:

`d/dxuv=u(dv)/dx+v(du)/dx`

`y=14tanxcosx + 10cscx`

Let's split the function into two parts and differentiate each separately.

`14tanxcosx`

`"Let"` `u = 14tanx` `"and"` `v=cosx`

`(du)/dx=14sec^2x`

`(dv)/dx=-sinx`

`d/dxuv=14cosx sec^2x-14tanxsinx`

`=14/cosx-(14sin^2x)/cosx=(14-14sin^2x)/cosx=(14(1-sin^2x))/cosx=(14cos^2x)/cosx=14cosx`

`d/dx10cscx=-10csc x cot x`

`dy/dx=14cosx-10csc x cot x`

Also note that we can simplify the first expression:

`d/dx 14 tanx cosx = d/dx 14 sinx/cosx cosx`
`" " = d/dx 14 sinx`
`" " = 14 cosx`