Tìm đạo hàm của các hàm số sau:
1, \(y=3^{(\dfrac{x}{\ln(x)})}\)
2, \(y=\dfrac{1}{2}tan^2(x)+\ln(tan(x))\)
3, 
\(y=\sqrt[3]{ln^2(2x)}\)
1.
\(y'=\left(\dfrac{x}{lnx}\right)'.3^{\dfrac{x}{lnx}}.ln3=\dfrac{lnx-1}{ln^2x}.3^{\dfrac{x}{lnx}}.ln3\)
2.
\(y'=\left(tanx\right)'.tanx+\left(tanx\right)'.\dfrac{1}{tanx}=\dfrac{tanx}{cos^2x}+\dfrac{1}{tanx.cos^2x}\)
3.
\(y=\left(ln2x\right)^{\dfrac{2}{3}}\Rightarrow y'=\left(ln2x\right)'.\dfrac{2}{3}.\left(ln2x\right)^{-\dfrac{1}{3}}=\dfrac{1}{3x\sqrt[3]{ln2x}}\)
Tính đạo hàm của các hàm số sau:
g) \(y = \ln (x^2+x+1)\)
l) \(y = \dfrac{\ln x}{x+1}\)
g: \(y=ln\left(x^2+x+1\right)\)
=>\(y'=\dfrac{\left(x^2+x+1\right)'}{x^2+x+1}=\dfrac{2x+1}{x^2+x+1}\)
l: \(y=\dfrac{lnx}{x+1}\)
=>\(y'=\dfrac{\left(lnx\right)'\cdot\left(x+1\right)-\left(x+1\right)'\left(lnx\right)}{\left(x+1\right)^2}\)
=>\(y'=\dfrac{\dfrac{1}{x}\left(x+1\right)-lnx}{\left(x+1\right)^2}\)
\(\Leftrightarrow y'=\dfrac{\dfrac{\left(x+1\right)}{x}-lnx}{\left(x+1\right)^2}\)
Tính đạo hàm của các hàm số sau :
a) \(y=\dfrac{1}{\left(2+3x\right)^2}\)
b) \(y=\sqrt[3]{\left(3x-2\right)^2};\left(x\ne\dfrac{2}{3}\right)\)
c) \(y=\dfrac{1}{\sqrt[3]{3x-7}}\)
d) \(y=3x^{-3}-\log_3x\)
e) \(y=\left(3x^2-2\right)\log_2x\)
g) \(y=\ln\left(\cos x\right)\)
h) \(y=e^x\sin x\)
i) \(y=\dfrac{e^x-e^{-x}}{x}\)
1. Tính đạo hàm của các hàm số sau:
a, \(y=\dfrac{2x-1}{x-1}\)
b, \(y=\dfrac{2x+1}{1-3x}\)
c, \(y=\dfrac{x^2+2x+2}{x+1}\)
d, \(y=\dfrac{2x^2}{x^2-2x-3}\)
e, \(y=x+1-\dfrac{2}{x-1}\)
g, \(y=\dfrac{2x^2-4x+5}{2x+1}\)
2. Tính đạo hàm của các hàm số sau:
a, \(y=\left(x^2+x+1\right)^4\)
b, y= (1-2x2)5
c, \(y=\left(\dfrac{2x+1}{x-1}\right)^3\)
d, \(y=\dfrac{\left(x+1\right)^2}{\left(x-1\right)^3}\)
e, \(y=\dfrac{1}{\left(x^2-2x+5\right)^2}\)
f, \(y=\left(3-2x^2\right)^4\)
a. \(y'=\dfrac{-1}{\left(x-1\right)}\)
b. \(y'=\dfrac{5}{\left(1-3x\right)^2}\)
c. \(y=\dfrac{\left(x+1\right)^2+1}{x+1}=x+1+\dfrac{1}{x+1}\Rightarrow y'=1-\dfrac{1}{\left(x+1\right)^2}=\dfrac{x^2+2x}{\left(x+1\right)^2}\)
d. \(y'=\dfrac{4x\left(x^2-2x-3\right)-2x^2\left(2x-2\right)}{\left(x^2-2x-3\right)^2}=\dfrac{-4x^2-12x}{\left(x^2-2x-3\right)^2}\)
e. \(y'=1+\dfrac{2}{\left(x-1\right)^2}=\dfrac{x^2-2x+3}{\left(x-1\right)^2}\)
g. \(y'=\dfrac{\left(4x-4\right)\left(2x+1\right)-2\left(2x^2-4x+5\right)}{\left(2x+1\right)^2}=\dfrac{4x^2+4x-14}{\left(2x+1\right)^2}\)
2.
a. \(y'=4\left(x^2+x+1\right)^3.\left(x^2+x+1\right)'=4\left(x^2+x+1\right)^3\left(2x+1\right)\)
b. \(y'=5\left(1-2x^2\right)^4.\left(1-2x^2\right)'=-20x\left(1-2x^2\right)^4\)
c. \(y'=3\left(\dfrac{2x+1}{x-1}\right)^2.\left(\dfrac{2x+1}{x-1}\right)'=3\left(\dfrac{2x+1}{x-1}\right)^2.\left(\dfrac{-3}{\left(x-1\right)^2}\right)=\dfrac{-9\left(2x+1\right)^2}{\left(x-1\right)^4}\)
d. \(y'=\dfrac{2\left(x+1\right)\left(x-1\right)^3-3\left(x-1\right)^2\left(x+1\right)^2}{\left(x-1\right)^6}=\dfrac{-x^2-6x-5}{\left(x-1\right)^4}\)
e. \(y'=-\dfrac{\left[\left(x^2-2x+5\right)^2\right]'}{\left(x^2-2x+5\right)^4}=-\dfrac{2\left(x^2-2x+5\right)\left(2x-2\right)}{\left(x^2-2x+5\right)^4}=-\dfrac{4\left(x-1\right)}{\left(x^2-2x+5\right)^3}\)
f. \(y'=4\left(3-2x^2\right)^3.\left(3-2x^2\right)'=-16x\left(3-2x^2\right)^3\)
Câu 28: Tính đạo hàm của hàm số y = 3x+1
A. y' = 3x+1ln3 B. y' = (1 + x) . 3x
C. y' = \(\dfrac{3^{x+1}}{ln3}\) D. Y' = \(\dfrac{3^{X+1}.ln3}{1+x}\)
Ta có:
\(y'=\left(3^{x+1}\right)'\)
\(=3^{x+1}ln3\)
\(\Rightarrow A\)
-Chúc bạn học tốt-
1. Đạo hàm của hàm số y= \(\left(x^3-5\right).\sqrt{x}\) bằng bao nhiêu?
2. Đạo hàm của hàm số y= \(\dfrac{1}{2}x^6-\dfrac{3}{x}+2\sqrt{x}\) là?
3. Hàm số y= \(2x+1+\dfrac{2}{x-2}\) có đạo hàm bằng?
1. \(y'=3x^2\sqrt{x}+\dfrac{x^3-5}{2\sqrt{x}}=\dfrac{7x^3-5}{2\sqrt{x}}\)
2. \(y'=3x^5+\dfrac{3}{x^2}+\dfrac{1}{\sqrt{x}}\)
3. \(y'=2-\dfrac{2}{\left(x-2\right)^2}\)
Đạo hàm của hàm số \(y=\left(x^2-\dfrac{2}{x}\right)^3\)là:
A. \(y'=6\left(x+\dfrac{1}{x^2}\right)\left(x^2-\dfrac{2}{x}\right)^2\)
B. \(y'=3\left(x^2-\dfrac{2}{x}\right)^2\)
C. \(y'=6\left(x-\dfrac{1}{x^2}\right)\left(x^2-\dfrac{2}{x}\right)^2\)
D. \(y'=6\left(x-\dfrac{1}{x}\right)\left(x^2-\dfrac{2}{x}\right)^2\)
\(y'=3\left(x^2-\dfrac{2}{x}\right)^2.\left(x^2-\dfrac{2}{x}\right)'=3\left(x^2-\dfrac{2}{x}\right)^2\left(2x+\dfrac{2}{x^2}\right)\)
\(=6\left(x+\dfrac{1}{x^2}\right)\left(x^2-\dfrac{2}{x}\right)^2\)
4. Tính đạo hàm của các hàm số sau:
a) \(y = (3x^2-4x+1)^{-4}\)
b) \(y = 3^{x^2-1} + e^{-x+1}\)
c) \(y = \ln (x^2-4x) + \log_{3} (2x-1)\)
d) \(y =x . \ln x + 2^{\frac{x-1}{x+1}}\)
e) \(y = x^{-7} - \ln (x^2-1)\)
`a)TXĐ:R\\{1;1/3}`
`y'=[-4(6x-4)]/[(3x^2-4x+1)^5]`
`b)TXĐ:R`
`y'=2x. 3^[x^2-1] ln 3-e^[-x+1]`
`c)TXĐ: (4;+oo)`
`y'=[2x-4]/[x^2-4x]+2/[(2x-1).ln 3]`
`d)TXĐ:(0;+oo)`
`y'=ln x+2/[(x+1)^2].2^[[x-1]/[x+1]].ln 2`
`e)TXĐ:(-oo;-1)uu(1;+oo)`
`y'=-7x^[-8]-[2x]/[x^2-1]`
Lời giải:
a.
$y'=-4(3x^2-4x+1)^{-5}(3x^2-4x+1)'$
$=-4(3x^2-4x+1)^{-5}(6x-4)$
$=-8(3x-2)(3x^2-4x+1)^{-5}$
b.
$y'=(3^{x^2-1})'+(e^{-x+1})'$
$=(x^2-1)'3^{x^2-1}\ln 3 + (-x+1)'e^{-x+1}$
$=2x.3^{x^2-1}.\ln 3 -e^{-x+1}$
c.
$y'=\frac{(x^2-4x)'}{x^2-4x}+\frac{(2x-1)'}{(2x-1)\ln 3}$
$=\frac{2x-4}{x^2-4x}+\frac{2}{(2x-1)\ln 3}$
d.
\(y'=(x\ln x)'+(2^{\frac{x-1}{x+1}})'=x(\ln x)'+x'\ln x+(\frac{x-1}{x+1})'.2^{\frac{x-1}{x+1}}\ln 2\)
\(=x.\frac{1}{x}+\ln x+\frac{2}{(x+1)^2}.2^{\frac{x-1}{x+1}}\ln 2\\ =1+\ln x+\frac{2^{\frac{2x}{x+1}}\ln 2}{(x+1)^2}\)
e.
\(y'=-7x^{-8}-\frac{(x^2-1)'}{x^2-1}=-7x^{-8}-\frac{2x}{x^2-1}\)
Cho hàm số \(y=\sqrt{x+\sqrt{x^2+1}}\). Tính đạo gàm của hàm số.
A. \(y'=\dfrac{x+\sqrt{x^2+1}}{2\sqrt{x^2+1}}\)
B. \(y'=\dfrac{\sqrt{x+\sqrt{x^2+1}}}{\sqrt{x^2+1}}\)
C. \(y'=\dfrac{\sqrt{x^2+1}}{2\sqrt{\sqrt{x+\sqrt{x^2+1}}}}\)
D. \(y'=\dfrac{\sqrt{x+\sqrt{x^2+1}}}{2\sqrt{x^2+1}}\)
\(y'=\dfrac{\left(x+\sqrt{x^2+1}\right)'}{2\sqrt{x+\sqrt{x^2+1}}}=\dfrac{1+\dfrac{x}{\sqrt{x^2+1}}}{2\sqrt{x+\sqrt{x^2+1}}}=\dfrac{x+\sqrt{x^2+1}}{2\sqrt{x^2+1}.\sqrt{x+\sqrt{x^2+1}}}\)
\(=\dfrac{\sqrt{x+\sqrt{x^2+1}}}{2\sqrt{x^2+1}}\)
tính đạo hàm của các hàm số sau
a) \(y=\dfrac{x^2+3x-1}{x+2}\)
b) \(y=\dfrac{2x^2-x}{x^2+1}\)
c) \(y=\dfrac{3-2x}{x-1}+\sqrt{2x-3}\)
a: \(y'=\dfrac{\left(x^2+3x-1\right)'\cdot\left(x+2\right)-\left(x^2+3x-1\right)\cdot\left(x+2\right)'}{\left(x+2\right)^2}\)
\(=\dfrac{\left(2x+3\right)\left(x+2\right)-\left(x^2+3x-1\right)}{\left(x+2\right)^2}\)
\(=\dfrac{2x^2+7x+6-x^2-3x+1}{\left(x+2\right)^2}=\dfrac{x^2+4x+7}{\left(x+2\right)^2}\)
b: \(y'=\dfrac{\left(2x^2-x\right)'\cdot\left(x^2+1\right)-\left(2x^2-x\right)\left(x^2+1\right)'}{\left(x^2+1\right)^2}\)
\(=\dfrac{4x\left(x^2+1\right)-2x\left(2x^2-x\right)}{\left(x^2+1\right)^2}\)
\(=\dfrac{4x^3+4x-4x^3+2x^2}{\left(x^2+1\right)^2}=\dfrac{2x^2+4x}{\left(x^2+1\right)^2}\)
c: \(\left(\dfrac{3-2x}{x-1}\right)'=\dfrac{\left(3-2x\right)'\left(x-1\right)-\left(3-2x\right)\left(x-1\right)'}{\left(x-1\right)^2}\)
\(=\dfrac{-2\left(x-1\right)-\left(3-2x\right)}{\left(x-1\right)^2}=\dfrac{-2x+2-3+2x}{\left(x-1\right)^2}=-\dfrac{1}{\left(x-1\right)^2}\)
\(\left(\sqrt{2x-3}\right)'=\dfrac{\left(2x-3\right)'}{2\sqrt{2x-3}}=\dfrac{1}{\sqrt{2x-3}}\)
\(y'=\left(\dfrac{3-2x}{x-1}\right)'+\left(\sqrt{2x-3}\right)'\)
\(=\dfrac{-1}{\left(x-1\right)^2}+\dfrac{1}{\sqrt{2x-3}}\)
