91Ó°ÊÓ

Differentiate with respect to \(x\) : (a) \(\ln \left\\{\frac{\cos x+\sin x}{\cos x-\sin x}\right\\}\) (b) \(\ln (\sec x+\tan x)\) (c) \(\sin ^{4} x \cos ^{3} x\)

Find \(\frac{\mathrm{d} y}{\mathrm{~d} x}\) when: (a) \(y=\frac{x \sin x}{1+\cos x}\) (b) \(y=\ln \left\\{\frac{1-x^{2}}{1+x^{2}}\right\\}\)

Write out the solutions carefully. They are all quite straightforward. If \(x^{2}+y^{2}-2 x+2 y=23\), find \(\frac{\mathrm{d} y}{\mathrm{~d} x}\) and \(\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}\) at the point where \(x=-2, y=3\)

If \(y\) is a function of \(x\), and \(x=\frac{e^{t}}{e^{t}+1}\) show that \(\frac{\mathrm{d} y}{\mathrm{~d} t}=x(1-x) \frac{\mathrm{d} y}{\mathrm{~d} x}\).

Write out the solutions carefully. They are all quite straightforward. Find an expression for \(\frac{\mathrm{d} y}{\mathrm{~d} x}\) when \(x^{3}+y^{3}+4 x y^{2}=5\).

Find \(\frac{\mathrm{d} y}{\mathrm{~d} x}\) when \(x^{3}+y^{3}-3 x y^{2}=8\).

Write out the solutions carefully. They are all quite straightforward. If \(x=3(1-\cos \theta)\) and \(y=3(\theta-\sin \theta)\) find \(\frac{\mathrm{d} y}{\mathrm{~d} x}\) and \(\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}\) in their simplest forms.

Differentiate: (a) \(y=e^{\sin ^{2} 5 x}\) (b) \(y=\ln \left\\{\frac{\cosh x-1}{\cosh x+1}\right\\}\) (c) \(y=\ln \left\\{e^{x}\left(\frac{x-2}{x+2}\right)^{3 / 4}\right\\}\)

If \((x-y)^{3}=A(x+y)\), prove that \((2 x+y) \frac{\mathrm{d} y}{\mathrm{~d} x}=x+2 y\)

If \(x^{2}-x y+y^{2}=7\), find \(\frac{\mathrm{d} y}{\mathrm{~d} x}\) and \(\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}\) at \(x=3, y=2\).