Question

Differentiate and simplify!! please help??

Answer 1

6(a) `(df)/(dx)=sqrt(x^2-1)/x+2xsec^(-1)x`
6(b) `(dg)/(dx)=-cotx`

Explanation:

6(a) As `f(x)=(x^2-1)sec^(-1)x` and `d/dx(sec^(-1)x)=1/sqrt(x^4-x^2)`

Using product formula `(df)/(dx)=(x^2-1)/sqrt(x^4-x^2)+2xsec^(-1)x`

= `(x^2-1)/(xsqrt(x^2-1))+2xsec^(-1)x`

= `sqrt(x^2-1)/x+2xsec^(-1)x`

6(b) `g(x)=ln(2cscx)`

Using chain rule `(dg)/(dx)=1/(2cscx)xx2(-cotxcscx)`

= `-sinxcosx/sinx xx1/sinx`

= `-cotx`