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

Answer 1

(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