Question

Differentiate and simplify! Please help!!?

Answer 1

(a) `(df)/(dx)=2cotxcscxe^(-2cscx)`
(b) `(df)/(dx)=6x^2sec^2(2x^3+5)`

Explanation:

When we have `x` in power, we take log of each side and use implicit differentiation.

As `y=f(x)=e^(-2cscx)`, we have `lny=-2cscx`

and taking differential of each side `1/y(dy)/(dx)=-2(-cotxcscx)`

or `(dy)/(dx)=2ycotxcscx=2cotxcscxe^(-2cscx)`

Additional information - Observe we could also have used function of function or chain rule too as `d/(dx)e^(f(x))=f'(x)e^(f(x)`, but this is done so that you can use it when we have something like `y=a^x`.

(b) As `f(x)=cot(2x^3+5)`, using chain rule

`f'(x)=sec^2(2x^3+5)xxd/(dx)(2x^3+5)`

= `sec^2(2x^3+5)xx6x^2=6x^2sec^2(2x^3+5)`