Bài 3 trang 71

Đề bài

Tìm đạo hàm của mỗi hàm số sau:

a) $y=4x^3-3x^2+2x+10$;

b) $y=\frac{x+1}{x-1}$;

c) $y=-2x\sqrt{x}$;

d) $y=3\sin x+4\cos x-\tan x$;

e) $y=4^x+2e^x$;

f) $y=x\ln x$.

Lời giải chi tiết

a) $y^{\prime }={\left(4x^3-3x^2+2x+10\right)}^{\prime }$

$=4.3x^{3-1}-3.2x^{2-1}+2x^{1-1}$

$=12x^2-6x+2$.

b) $y^{\prime }={\left(\frac{x+1}{x-1}\right)}^{\prime }$

$=\frac{(x+1{)}^{\prime }(x-1)-(x+1\left)\right(x-1{)}^{\prime }}{{(x-1)}^2}$

$=\frac{1.(x-1)-(x+1).1}{{(x-1)}^2}$

$=\frac{x-1-x-1}{{(x-1)}^2}=\frac{-2}{{(x-1)}^2}$.

c) $y^{\prime }={\left(-2x\sqrt{x}\right)}^{\prime }=-2{\left(x\sqrt{x}\right)}^{\prime }$

$=-2\left(1.\sqrt{x}+x.\frac{1}{2\sqrt{x}}\right)$

$=-2\left(\sqrt{x}+\frac{x}{2\sqrt{x}}\right)$

$=-2\left(\frac{2x}{2\sqrt{x}}+\frac{x}{2\sqrt{x}}\right)$

$=-\frac{3x}{\sqrt{x}}=-3\sqrt{x}$.

d) $y^{\prime }={\left(3\sin x+4\cos x-\tan x\right)}^{\prime }$

$=3\cos x-4\sin x+\frac{1}{{\cos }^{2}x}$

$=\frac{3{\cos }^{3}x-4\sin x.{\cos }^{2}x+1}{{\cos }^{2}x}$.

e) $y^{\prime }={\left(4^x+2e^x\right)}^{\prime }=4^x.\ln 4+2e^x$.

f) $y^{\prime }={\left(x\ln x\right)}^{\prime }=x^{\prime }\ln x+x{\left(\ln x\right)}^{\prime }$

$=\ln x+x.\frac{1}{x}=\ln x+1$.