Find $\frac{\mathrm{d} }{\mathrm{d} x}x\sin \left ( \sqrt{3x^{2}+5} \right )$ without using the chain rule. $$\frac{\mathrm{d} }{\mathrm{d} x}x\sin \left ( \sqrt{3x^{2}+5} \right )$$
I can't for the life of me differentiate this function while only using Trig Identities, Basic differentiation rules, and Limits (no L'Hopital, either.)

$$f'=\lim_{t \to 0}\frac{(x+t)\sin\sqrt{3(x+t)^2+5}-(x)\sin\sqrt{3(x)^2+5}}{t}$$ separate it into two limitation by $\pm(x)\sin\sqrt{3(x+t)^2+5}$
this means
$$f'=\lim_{t \to 0}\frac{(x+t)\sin\sqrt{3(x+t)^2+5}-(x)\sin\sqrt{3(x)^2+5}}{t}=\\
=\lim_{t \to 0}\frac{(x+t)\sin\sqrt{3(x+t)^2+5}\pm(x)\sin\sqrt{3(x+t)^2+5}-(x)\sin\sqrt{3(x)^2+5}}{t}\\
=\lim_{t \to 0}\frac{(x+t)\sin\sqrt{3(x+t)^2+5}-(x)\sin\sqrt{3(x+t)^2+5}}{t}+\lim_{t \to 0}\frac{+x\sin\sqrt{3(x+t)^2+5}-(x)\sin\sqrt{3(x)^2+5}}{t}$$
