Differentiate the functions trigonometry and derivatives

Differentiate $ln(\sec x+ \tan x)$ and $(\sin x)^3\cos 3x+ (\cos x)^3\sin 3x$ with respect to $x$, simplifying where possible.

Find the first and second derivatives (with respect to x) of the function $x= \sin(t^2+t+2)$ $y= 5^t$

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Welcome to MSE! Do you have thoughts on the problem and can share what you have tried? Also, it helps readability to format questions using MahtJax (see FAQ). Lastly, only one question per posting is preferred. Regards –  Amzoti Nov 18 '13 at 15:39

Using $$\sin3x=3\sin x-4\sin^3x,\cos3x=4\cos^3x-3\cos x$$
$$\sin^3x\cos3x+\cos^3x\sin3x$$ $$=\frac{(3\sin x-\sin3x)\cos3x+(\cos3x+3\cos x)\sin3x}4=\frac34(\sin x\cos3x+\cos x\sin3x)=\frac34\sin(x+3x)$$
@DK.H, use en.wikipedia.org/wiki/Chain_rule for the $\ln$ and the last $\sin$ function –  lab bhattacharjee Nov 18 '13 at 16:00