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.

\(y=\left(ln2x\right)^{\dfrac{2}{3}}\Rightarrow y'=\left(ln2x\right)'.\dfrac{2}{3}.\left(ln2x\right)^{-\dfrac{1}{3}}=\dfrac{1}{3x\sqrt[3]{ln2x}}\)

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. \(y'=2-\dfrac{2}{\left(x-2\right)^2}\)

1) \(f\left(x\right)=2x-5\)

\(f'\left(x\right)=2\)

\(\Rightarrow f'\left(4\right)=2\)

2) \(y=x^2-3\sqrt[]{x}+\dfrac{1}{x}\)

\(\Rightarrow y'=2x-\dfrac{3}{2\sqrt[]{x}}-\dfrac{1}{x^2}\)

3) \(f\left(x\right)=\dfrac{x+9}{x+3}+4\sqrt[]{x}\)

\(\Rightarrow f'\left(x\right)=\dfrac{1.\left(x+3\right)-1.\left(x+9\right)}{\left(x-3\right)^2}+\dfrac{4}{2\sqrt[]{x}}\)

\(\Rightarrow f'\left(x\right)=\dfrac{x+3-x-9}{\left(x-3\right)^2}+\dfrac{2}{\sqrt[]{x}}\)

\(\Rightarrow f'\left(x\right)=\dfrac{12}{\left(x-3\right)^2}+\dfrac{2}{\sqrt[]{x}}\)

\(\Rightarrow f'\left(x\right)=2\left[\dfrac{6}{\left(x-3\right)^2}+\dfrac{1}{\sqrt[]{x}}\right]\)

\(\Rightarrow f'\left(1\right)=2\left[\dfrac{6}{\left(1-3\right)^2}+\dfrac{1}{\sqrt[]{1}}\right]=2\left(\dfrac{3}{2}+1\right)=2.\dfrac{5}{2}=5\)

tham khảo:

a)\(y'=\dfrac{d}{dx}\left(x^3\right)-\dfrac{d}{dx}\left(3x^2\right)+\dfrac{d}{dx}\left(2x\right)+\dfrac{d}{dx}\left(1\right)\)

\(y'=3x^2-6x+2\)

b)\(\dfrac{d}{dx}\left(x^n\right)=nx^{n-1}\)

\(\dfrac{d}{dx}\left(\sqrt{x}\right)=\dfrac{1}{2\sqrt{x}}\)

\(\dfrac{d}{dx}\left(f\left(x\right)+g\left(x\right)\right)=f'\left(x\right)+g'\left(x\right)\)

\(\dfrac{d}{dx}\left(cf\left(x\right)\right)=cf'\left(x\right)\)

\(y'=\dfrac{d}{dx}\left(x^2\right)-\dfrac{d}{dx}\left(4\sqrt{x}\right)+\dfrac{d}{dx}\left(3\right)\)

\(y'=2x-2\sqrt{x}\)

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: \(\left(\dfrac{3-2x}{x-1}\right)'=\dfrac{\left(3-2x\right)'\left(x-1\right)-\left(3-2x\right)\left(x-1\right)'}{\left(x-1\right)^2}\)

\(=\dfrac{-2\left(x-1\right)-\left(3-2x\right)}{\left(x-1\right)^2}=\dfrac{-2x+2-3+2x}{\left(x-1\right)^2}=-\dfrac{1}{\left(x-1\right)^2}\)

\(\left(\sqrt{2x-3}\right)'=\dfrac{\left(2x-3\right)'}{2\sqrt{2x-3}}=\dfrac{1}{\sqrt{2x-3}}\)

\(y'=\left(\dfrac{3-2x}{x-1}\right)'+\left(\sqrt{2x-3}\right)'\)

\(=\dfrac{-1}{\left(x-1\right)^2}+\dfrac{1}{\sqrt{2x-3}}\)

a: \(y'=4\cdot3x^2-3\cdot2x+2=12x^2-6x+2\)

b: \(y'=\dfrac{\left(x+1\right)'\left(x-1\right)-\left(x+1\right)\left(x-1\right)'}{\left(x-1\right)^2}=\dfrac{x-1-x-1}{\left(x-1\right)^2}=\dfrac{-2}{\left(x-1\right)^2}\)

c: \(y'=-2\cdot\left(\sqrt{x}\cdot x\right)'\)

\(=-2\cdot\left(\dfrac{x+x}{2\sqrt{x}}\right)=-2\cdot\dfrac{2x}{2\sqrt{x}}=-2\sqrt{x}\)

d: \(y'=\left(3sinx+4cosx-tanx\right)\)'

\(=3cosx-4sinx+\dfrac{1}{cos^2x}\)

e: \(y'=\left(4^x+2e^x\right)'\)

\(=4^x\cdot ln4+2\cdot e^x\)

f: \(y'=\left(x\cdot lnx\right)'=lnx+1\)

a: \(y'=\left(x^2\right)'+\left(3x\right)'-\left(6x^6\right)'+\left(\dfrac{2x-3}{x-1}\right)'\)

\(=2x+3-6\cdot6x^5+\dfrac{\left(2x-3\right)'\left(x-1\right)-\left(2x-3\right)\left(x-1\right)'}{\left(x-1\right)^2}\)

\(=-36x^5+2x+3+\dfrac{2\left(x-1\right)-2x+3}{\left(x-1\right)^2}\)

\(=-36x^5+2x+3+\dfrac{1}{\left(x-1\right)^2}\)

b: \(\left(\sqrt{2x^2-3x+1}\right)'=\dfrac{\left(2x^2-3x+1\right)'}{2\sqrt{2x^2-3x+1}}\)

\(=\dfrac{4x-3}{2\sqrt{2x^2-3x+1}}\)

\(y'=3\cdot2x-4+\dfrac{4x-3}{2\sqrt{2x^2-3x+1}}\)

\(=6x-4+\dfrac{4x-3}{2\sqrt{2x^2-3x+1}}\)

c: \(\left(\sqrt{4x^2-3x+1}\right)'=\dfrac{\left(4x^2-3x+1\right)'}{2\sqrt{4x^2-3x+1}}\)

\(=\dfrac{8x-3}{2\sqrt{4x^2-3x+1}}\)

\(y'=\left(\sqrt{4x^2-3x+1}\right)'-4'=\dfrac{8x-3}{2\sqrt{4x^2-3x+1}}\)

\(a,y'=\left[\left(2x-3\right)^{10}\right]'\\ =10\left(2x-3\right)^9\left(2x-3\right)'\\ =20\left(2x-3\right)^9\\ b,y'=\left(\sqrt{1-x^2}\right)'\\ =\dfrac{\left(1-x^2\right)'}{2\sqrt{1-x^2}}\\ =-\dfrac{2x}{2\sqrt{1-x^2}}\\ =-\dfrac{x}{\sqrt{1-x^2}}\)