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\(y'=\frac{2x}{x^2+1}+\frac{2x-1}{\left(x^2-x+1\right)\ln2}\)
\(y=\log_2\left(\frac{x-4}{x+4}\right)\Rightarrow y'=\frac{\frac{8}{\left(x+4\right)^2}}{\left(\frac{x-4}{x+4}\right)\ln2}=\frac{8}{\left(x^2-16\right)\ln2}\)
\(y'=\frac{e^x}{2\sqrt{e^x}}+3.e^{3x-1}-\left(-\sin x+\cos x\right)5^{\sin x+\cos x}\ln5\)
\(=\frac{\sqrt{e^x}}{2}+3e^{3x-1}+\left(\sin x+\cos x\right).5^{\sin x+\cos x}\ln5\)
xét hàm số y=ln(\(x+\sqrt{1+x^2}\))
Ta có
y'=\(\frac{1}{x+\sqrt{1+x^2}}\left(1+\frac{x}{\sqrt{1+x^2}}\right)=\frac{1}{x+\sqrt{1+x^2}}.\frac{x+\sqrt{1+x^2}}{\sqrt{1+x^2}}=\frac{1}{\sqrt{1+x^2}}\)
\(\Rightarrow y'=\frac{2\left(\ln x\right)\frac{1}{x}}{3\sqrt[3]{\ln^4x}}=\frac{2}{3x\sqrt[3]{\ln x}}\)
a) Cách 1: y' = (9 -2x)'(2x3- 9x2 +1) +(9 -2x)(2x3- 9x2 +1)' = -2(2x3- 9x2 +1) +(9 -2x)(6x2 -18x) = -16x3 +108x2 -162x -2.
Cách 2: y = -4x4 +36x3 -81x2 -2x +9, do đó
y' = -16x3 +108x2 -162x -2.
b) y' = .(7x -3) +
(7x -3)'=
(7x -3) +7
.
c) y' = (x -2)'√(x2 +1) + (x -2)(√x2 +1)' = √(x2 +1) + (x -2) = √(x2 +1) + (x -2)
= √(x2 +1) +
=
.
d) y' = 2tanx.(tanx)' - (x2)' =
.
e) y' = sin
=
sin
.
\(L=\lim\limits_{x\rightarrow0}\frac{e^x-1}{\sqrt{x+1}-1}=\lim\limits_{x\rightarrow0}\frac{\left(e^x-1\right)\left(\sqrt{x+1}-1\right)}{x}=\lim\limits_{x\rightarrow0}\left[\frac{e^x-1}{x}.\left(\sqrt{x+1}-1\right)\right]=1.0=0\)
\(y=\log\left(\frac{1-\sqrt{x}}{2\sqrt{x}}\right)\Rightarrow y'=\frac{\left(\frac{1-\sqrt{x}}{2\sqrt{x}}\right)'}{\frac{1-\sqrt{x}}{x^2}\ln10}=\frac{-\frac{1}{2\sqrt{x}}.2\sqrt{x}-\frac{1}{\sqrt{x}}.\left(1-\sqrt{x}\right)}{\frac{1-\sqrt{x}}{2\sqrt{x}}\ln10}\)
\(=\frac{-1-\frac{1-\sqrt{x}}{\sqrt{x}}}{4x.\frac{1-\sqrt{x}}{2\sqrt{x}}\ln10}=\frac{1}{2x\left(\sqrt{x}-1\right)\ln10}\)