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Question
Fill in the blank:
Differentiate tan x and sec x w.r.t.x. using the formulae for differentiation of `"u"/"v" and 1/"v"` respectively
Solution
(i) Let y = `tan x = sin x/cos x`
∴ `("d"y)/("d"x) = "d"/("d"x) ((sin x)/(cos x))`
= `((cos x) "d"/("d"x) (sin x) - (sin x) "d"/("d"x) (cos x))/(cos^2x)`
= `((cos x)(cos x) - (sin x)(- sin x))/(cos^2x)`
= `(cos^2x + sin^2x)/(cos^2x)`
= `1/(cos^2x)`
= sec2x
(ii) Let y = sec x = `1/cos x`
∴ `("d"y)/("d"x) = "d"/("d"x) [1/cos x]`
= `((cos x) "d"/("d"x) (1) - (1) "d"/("d"x) (cos x))/(cos^2x)`
= `((cos x)(0) - (- sin x))/(cos^2x)`
= `sinx/cos^2x`
= `(1/(cos x))((sin x)/(cos x))`
= sec x tan x
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