Question

Differentiate `secx` using the definition of differential?

Answer 1

`d/(dx)secx=secxtanx`

Explanation:

For a function `f(x)`, its differential `(df)/(dx)=Lt_(h->0)(f(x+h)-f(x))/h`

Hence for `secx`, `d/(dx)secx=Lt_(h->0)(sec(x+h)-secx)/h`

= `Lt_(h->0)(1/cos(x+h)-1/cosx)/h`

= `Lt_(h->0)(cosx-cos(x+h))/(hcos(x+h)cosx)`

= `Lt_(h->0)(2sin((x+h+x)/2)sin((x+h-x)/2))/(hcos(x+h)cosx)`

= `Lt_(h->0)(sin(x+h/2)sin(h/2))/(h/2cos(x+h)cosx)`

= `Lt_(h->0)sin(x+h/2)/(cos(x+h)cosx)xxLt_(h->0)(sin(h/2))/(h/2)`xx

= `sinx/(cosxcosx) xx1`

= `1/cosx xxsinx/cosx`

= `secxtanx`