Question 28.7.5: If y=cos² x², find dy/dx....
If y=\cos ^{2} x^{2} , find \frac{d y}{d x} .
Step-by-Step
y=\left(\cos x^{2}\right)^{2} . Put x^{2}=t and \cos x^{2}=\cos t=u , so that
y=u^{2}, u=\cos t \text { and } t=x^{2} .\\\therefore \quad \frac{d y}{d u}=2 u, \frac{d u}{d t}=-\sin t \text { and } \frac{d t}{d x}=2 x \text {. }
so, \frac{d y}{d x}=\left(\frac{d y}{d u} \times \frac{d u}{d t} \times \frac{d t}{d x}\right)\\ \\\begin{array}{l}=-4 u x \sin t=-4 x \sin t \cos t \quad[\because u=\cos t]\\ \\=-4 x \sin x^{2} \cos x^{2}=-2 x \sin \left(2 x^{2}\right) \quad\left[\because t=x^{2}\right] .\end{array}
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