xét hàm số y=\(\sqrt{x+\sqrt{x}}+\sqrt{x}\) . ta có

y'=\(\frac{\left(x+\sqrt{x}\right)}{2\sqrt{x+\sqrt{x}}}+\frac{1}{2\sqrt{x}}=\frac{1+\frac{1}{2\sqrt{x}}}{2\sqrt{x+\sqrt{x}}}+\frac{1}{2\sqrt{x}}\)

=\(\frac{1+2\sqrt{x}}{4\sqrt{x}\sqrt{x+\sqrt{x}}}+\frac{1}{2\sqrt{x}}=\frac{1+2\sqrt{x}+2\sqrt{x+\sqrt{x}}}{4\sqrt{x}\sqrt{x+\sqrt{x}}}\)

a) y' = 2x - = 2x - .

b) y' = = .

c) y' = = = = .

d) y' = = = = .

xét hàm số y=ln(\(x+\sqrt{1+x^2}\))

Ta có

y'=\(\frac{1}{x+\sqrt{1+x^2}}\left(1+\frac{x}{\sqrt{1+x^2}}\right)=\frac{1}{x+\sqrt{1+x^2}}.\frac{x+\sqrt{1+x^2}}{\sqrt{1+x^2}}=\frac{1}{\sqrt{1+x^2}}\)

\(y=\log\left(\frac{1-\sqrt{x}}{2\sqrt{x}}\right)\Rightarrow y'=\frac{\left(\frac{1-\sqrt{x}}{2\sqrt{x}}\right)'}{\frac{1-\sqrt{x}}{x^2}\ln10}=\frac{-\frac{1}{2\sqrt{x}}.2\sqrt{x}-\frac{1}{\sqrt{x}}.\left(1-\sqrt{x}\right)}{\frac{1-\sqrt{x}}{2\sqrt{x}}\ln10}\)

                                 \(=\frac{-1-\frac{1-\sqrt{x}}{\sqrt{x}}}{4x.\frac{1-\sqrt{x}}{2\sqrt{x}}\ln10}=\frac{1}{2x\left(\sqrt{x}-1\right)\ln10}\)

\(y'=\frac{e^x}{2\sqrt{e^x}}+3.e^{3x-1}-\left(-\sin x+\cos x\right)5^{\sin x+\cos x}\ln5\)

    \(=\frac{\sqrt{e^x}}{2}+3e^{3x-1}+\left(\sin x+\cos x\right).5^{\sin x+\cos x}\ln5\)