Differentiation — Maths STD 12 Science — Question

Differentiating w.r.t. x, we get

$\begin{aligned} & \frac{d y}{d x}=\frac{d}{d x}\left[e^{3 \sin ^2 x-2 \cos ^2 x}\right] \\ & =e^{3 \sin ^2 x-2 \cos ^2 x} \cdot \frac{d}{d x}\left(3 \sin ^2 x-2 \cos ^2 x\right) \\ & =e^{3 \sin ^2 x-2 \cos ^2 x} \cdot\left[3 \frac{d}{d x}(\sin x)^2-2 \frac{d}{d x}(\cos x)^2\right] \\ & =e^{3 \sin ^2 x-2 \cos ^2 x} \cdot\left[3 \times 2 \sin x \cdot \frac{d}{d x}(\sin x)-2 \times 2 \cos x \cdot \frac{d}{d x}(\cos x)\right] \\ & =e^{3 \sin ^2 x-2 \cos ^2 x} \cdot[6 \sin x \cos x-4 \cos x(-\sin x)] \\ & =e^{3 \sin ^2 x-2 \cos ^2 x} \cdot(10 \sin x \cos x) \\ & =5\left(2 \sin x \cos ^2 x\right) \cdot e^{3 \sin 2 x-2 \cos ^2 x} \\ & =5 \sin 2 x \cdot e^{3 \sin ^2 x-2 \cos ^2 x} .\end{aligned}$