1.
\(f'\left(x\right)=\dfrac{1}{cos^2x}\) ; \(g'\left(x\right)=-\dfrac{1}{1-x}\)
\(\Rightarrow f'\left(0\right)=1\) ; \(g'\left(0\right)=-1\)
\(\Rightarrow\dfrac{f'\left(0\right)}{g'\left(0\right)}=-1\)
2.
\(y'=-3x^2\)
\(k=y'\left(-1\right)=-3.\left(-1\right)^2=-3\)
3.
\(y'=3x^2-3\)
\(k=y'\left(x_0\right)=9\Rightarrow3x_0^2-3=9\)
\(\Rightarrow x_0^2=4\)
\(\Rightarrow x_0=\pm2\)
7.
\(y'=\dfrac{2\left(1-4x\right)-\left(-4\right).\left(2x+3\right)}{\left(1-4x\right)^2}=\dfrac{14}{\left(1-4x\right)^2}\)
8.
\(y'=\dfrac{x-1-\left(x+1\right)}{\left(x-1\right)^2}=\dfrac{-2}{\left(x-1\right)^2}\)
\(y'\left(2\right)=\dfrac{-2}{\left(2-1\right)^2}=-2\)
9.
\(y'=\dfrac{\left(-2x+2\right)\left(x-1\right)-\left(-x^2+2x-2\right)}{\left(x-1\right)^2}\)
\(=\dfrac{-x^2+2x}{\left(x-1\right)^2}\)
4.
\(y'=3x^2\)
\(y'\left(-1\right)=3\)
Pt tiếp tuyến:
\(y=3\left(x+1\right)-1=3x+2\)
5.
\(y'=2cosx.e^{2sinx}\)
\(y'\left(\dfrac{\pi}{6}\right)=2cos\left(\dfrac{\pi}{6}\right).e^{2sin\dfrac{\pi}{6}}=\sqrt{3}e\)
6.
\(y'=2-\dfrac{4}{\left(x-1\right)^2}\)
\(k=y'\left(3\right)=2-\dfrac{4}{\left(3-1\right)^2}=1\)
10.
\(y'=3\left(x^2-\dfrac{1}{x}\right)^2.\left(x^2-\dfrac{1}{x}\right)'=3\left(x^2-\dfrac{1}{x}\right)^2\left(2x+\dfrac{1}{x^2}\right)\)
\(=\dfrac{3\left(x^3-1\right)^2\left(3x^3+1\right)}{x^4}\)
11.
\(y'=2\left(x^4+2x^2+2\right).\left(x^4+2x^2+2\right)'\)
\(=2\left(x^4+2x^2+2\right)\left(4x^3+4x\right)\)
\(\Rightarrow y'\left(0\right)=2.\left(0+0+2\right).\left(0+0\right)=0\)
12.
\(y=x^2+\dfrac{x^2\left(x-3\right)}{x-3}=2x^2\)
\(\Rightarrow y'=4x\)
13.
D đúng, cả 3 công thức đều đúng
13.1
\(f'\left(x\right)=2x\)
\(g'\left(x\right)=4+\pi.cos\left(\dfrac{\pi x}{2}\right)\)
\(\Rightarrow f'\left(1\right)=2\) ; \(g'\left(1\right)=4+\pi.cos\left(\dfrac{\pi}{2}\right)=4\)
\(\Rightarrow\dfrac{f'\left(1\right)}{g'\left(1\right)}=\dfrac{1}{2}\)
14.
\(y'=5cos^44x.\left(cos4x\right)'=-20cos^4x.sin4x\) (D)
15.
\(\left(sinx\right)'=cosx\), B sai
16.
\(y'=-\dfrac{5}{x^2}\)
\(y''=\dfrac{10}{x^3}\)
\(M=x.y''+2y'=x.\dfrac{10}{x^3}+2.\left(-\dfrac{5}{x^2}\right)=\dfrac{10}{x^2}-\dfrac{10}{x^2}=0\)
17.
\(y'=\dfrac{e^x+e^{-x}}{2}\Rightarrow f'\left(1\right)=\dfrac{e+\dfrac{1}{e}}{2}=\dfrac{e^2+1}{2e}\)
18.
\(y'=e^{cosx}.\left(cosx\right)'=-sinx.e^{cosx}\)
\(y'\left(\dfrac{\pi}{2}\right)=-sin\left(\dfrac{\pi}{2}\right).e^{cos\dfrac{\pi}{2}}=-1.e^0=-1\)
19.
\(y'=cosx.e^{cosx}+\left(-sinx.e^{cosx}\right).sinx\)
\(=e^{cosx}\left(cosx-sin^2x\right)\)
\(y'\left(\dfrac{\pi}{2}\right)=e^{cos\dfrac{\pi}{2}}\left(cos\dfrac{\pi}{2}-sin^2\dfrac{\pi}{2}\right)=1.\left(0-1\right)=-1\)
20.
\(y'=e^{sin^2x}.\left(sin^2x\right)'=e^{sin^2x}.2sinx.cosx=e^{sin^2x}.sin2x\)
21.
\(f'\left(x\right)=\dfrac{1}{x}\Rightarrow f'\left(\dfrac{1}{e}\right)=e\)
22.
\(y'=lnx+\dfrac{x}{x}-1=lnx\)
23.
\(y'=\dfrac{1}{sinx}.\left(sinx\right)'=\dfrac{cosx}{sinx}=cotgx\)
