whats the differentiaion of sin^2(x)?

Der ( u^2 ) = 2u. Der(u)

Hence

Der( sin^2 x )

= 2 sin x . Der( sin x)

= 2 sin x cos x

= sin( 2x ). ..........................Ans.

Chain rule

d/dx sin² x = 2 · sin x · cos x

Chain rule states:

d/dx f( g(x) ) = f'( g(x) ) · g'(x)

In the case of sin² x

f(x) = x²

g(x) = sin x

So, f( g(x) ) = sin² x

d/dx sin² x = d/dx f( g(x) ) = f'( g(x) ) · g'(x)

d/dx f(x) = d/dx x² = 2x

d/dx g(x) = d/dx sin x = cos x

d/dx sin² x = 2 sin x cos x or sin2x

derivative of u² = 2u du

u = sin x

du = cos x

⇔

f (x) = sin² x

f '(x) = 2 u du = 2 sin x cos x = sin 2x ⇦ answer

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method 1

let sinx = u then

dy/dx=(dy/du)( du/dx)

=2sinxcosx

=sin2x

method 2

use identities

sin^2 (x)= (sinx)^2= (1-cos2x)/2 = (1/2)-(cos2x)/2

so differentiation:

(1/2) is constant becomes 0 when differentiated

(cos2x)/2 becomes sin2x