Differentiation...

In a combination, the ordering of the selected objects is immaterial whereas in a permutation, the ordering is essential.

If $$y={(\tan{x})}^{\displaystyle{(\tan{x})}^{\displaystyle\tan{x}}}$$, then find $$\displaystyle\frac{dy}{dx}$$ at $$\displaystyle x=\frac{\pi}{4}$$.

There are $$6$$ boxes numbered $$1, 2 ....... 6$$. Each box is to be filled up either with a red or a green ball in such a way that at least $$1$$ box contains a green ball and the boxes containing green balls are consecutively numbered. The total number of ways in which this can be done is:

If $$\displaystyle y=5^{3-x^2}+(3-x^2)^5$$, then $$\displaystyle \frac{dy}{dx}=$$

If $$y=\log_{3}x+3 \log_{e} x+2 \tan x$$, then $$\displaystyle \frac{dy}{dx}=$$

If $$\displaystyle y=e^{x \log a}+e^{a \log x}+e^{a \log a}$$, then $$\displaystyle \frac{dy}{dx}=$$

If $$f'(x)=\sin x+\sin 4x\cdot \cos x$$, then $$f'\left (2x^2+\displaystyle \frac {\pi}{2}\right )$$ is

If $$y=|\cos x|+|\sin x|$$, then $$\displaystyle \dfrac {dy}{dx}$$ at $$x=\dfrac {2\pi}{3}$$ is

Find the derivative of $$\sec^{-1}\left (\displaystyle \frac {x+1}{x-1}\right )+\sin^{-1}\left (\displaystyle \frac {x-1}{x+1}\right )$$

If $$y=\log_{10}x+\log_x 10+\log_xx+\log_{10} 10$$, then $$\displaystyle \frac{dy}{dx}=$$