Derivative of tan, cot, sec, cosec and e^-x.

df/dx = = [dg/dx]h + g[dh/dx] when f =gh
df/dx = = {[dg/dx]h - g[dh/dx]}/h2 when h≠0 and f = g/h

Derivative of tan x

tan x = sin x /cos x

d(tan x)/dx
= d(sin x/cos x)/dx
= [d(sin x)/dx cos x - sin x d(cos x)/dx]/cos2 x
= [cos x cos x - sin x (- sin x)]/cos2 x
= [cos2 x + sin2 x]/cos2 x
= 1/cos2 x = sec2 x


Derivative of cot x

cot x = cos x /sin x

d(cot x)/dx
= d(cos x/sin x)/dx
= [d(cos x)/dx sin x - cos x d(sin x)/dx]/sin2 x
= [- sin x sin x - cos x cos x ]/sin2 x
= -[sin2 x + cos2 x]/sin2 x
= -1/sin2 x = -cosec2 x


Derivative of sec x

sec x = 1/cos x

d(sec x)/dx
= d(1/cos x)/dx
= [d(1)/dx cos x - 1 d(cos x)/dx]/cos2 x
= [ sin x ]/cos2 x
= sec x tan x


Derivative of cosec x

cosec x = 1/sin x

d(cosec x)/dx
= d(1/sin x)/dx
= [d(1)/dx sin x - 1 d(sin x)/dx]/sin2 x
= [ -cos x ]/sin2 x
= -cosec x cot x

Derivative of e-x

e-x = 1/ex

d(e-x)/dx
= d(1/ex)/dx
= [d(1)/dx ex - 1 d(ex)/dx]/e2x
= [ -ex ]/e2x
= -e-x




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