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Question
Find the derivative of the following function:
sec x
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Solution
Let f(x) = sec x
∴ f(x + h) = sec (x + h)
f(x + h) − f(x) = sec (x + h) − sec x
= `1/(cos(x + h)) - 1/(cos x)`
= `(cos x - cos (x + h))/(cos (x + h) cos x)`
= `(2sin (x + h/2)sin (h/2))/(cos (x + h) cos x`
`f'(x) = lim_(h → 0)(f(x + h) - f(x))/h`
= `lim_(h → 0) (2 sin(x + h/2) sin (h/2))/(h cos (x + h) cos x)`
= `lim_(h → 0) (sin (x + h/2))/(cos (x + h)cos x). ((sin h/2)/(h/2))`
= `(sin x)/(cos x. cos x)` ......`[∵ lim_(h → 0)((sin h/2)/(h/2)) = 1]`
= `1/(cos x). (sin x)/(cos x)`
= sec x tan x
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