Question

Differentiate the following w.r.t.x:

sin²x² – cos²x² 

Answer 1

Let y = sin²x² – cos²x² 
Differentiating w.r.t. x, we get
${\displaystyle \frac{\text{dy}}{\text{dx}}=\frac{\text{d}}{\text{dx}}[{\sin }^2{x}^2-{\cos }^2{x}^2]}$

= ${\displaystyle \frac{\text{d}}{\text{dx}}{\left({\sin x}^2\right)}^2-\frac{\text{d}}{\text{dx}}{\left({\cos x}^2\right)}^2}$

= ${\displaystyle 2{\sin x}^2.\frac{\text{d}}{\text{dx}}\left({\sin x}^2\right)-2{\cos x}^2.\frac{\text{d}}{\text{dx}}\left({\cos x}^2\right)}$

= ${\displaystyle 2{\sin x}^2.{\cos x}^2.\frac{\text{d}}{\text{dx}}\left(x^2\right)-2{\cos x}^2.(-{\sin x}^2).\frac{\text{d}}{\text{dx}}\left(x^2\right)}$

= 2sin x² . cos x² x 2x + 2sinx² . cosx² x 2x
= 4x (2sinx² . cosx²)
= 4x sin(2x²).