Tính đạo hàm của hàm số sau
a) y= 32x^100+ 32x^50-32x²+32-32
b) y= 32x^100+ 32x^50-32x² – 32√x + 32x – 32
c) f(x) = (x^10 – 32√x+32) (x² – 32x+32)
d) y= 32x – 10 / 32x + 20
e) y= 10x + 32 /20x – 32
f) y = (32x^20 – 32x^7 + 32x – 32)^10
g) y= sin √32x² -32x +32
Đáp án:
$a) y’=3200x^{99}+1600x^{49}-64x\\
b) y’=3200x^{99}+1600x^{49}-64x-\frac{16}{\sqrt{x}}+32\\
c) f'(x)=10x^9-\frac{16}{\sqrt{x}}(x^2 – 32x+32)+(x^{10} – 32\sqrt{x}+32) (2x-32)\\
d) y’=\frac{320}{(32x+20)^2}\\
e) y’=\frac{-960}{(20x-32)^2}\\
f) y’=10.(32x^{20} – 32x^7 + 32x – 32)^9.(640x^{19} – 224x^6 + 32)\\
g) y’=\frac{(32x -16)}{\sqrt{32x^2 -32x +32} }cos \sqrt{32x^2 -32x +32}\\$
Giải thích các bước giải:
$a) y= 32x^{100}+ 32x^{50}-32x^2+32-32\\
\Rightarrow y’=\left ( 32x^{100}+ 32x^{50}-32x^2 \right )’\\
=32.100x^{99}+32.50x^{49}-32.2x\\
=3200x^{99}+1600x^{49}-64x\\
b) y= 32x^{100}+ 32x^{50}-32x^2 – 32\sqrt{x} + 32x – 32\\
\Rightarrow y’=\left (32x^{100}+ 32x^{50}-32x^2 – 32\sqrt{x} + 32x – 32 \right )\\
=32.100.x^{99}+32.50x^{49}-32.2x-32.\frac{1}{2\sqrt{x}}+32\\
=3200x^{99}+1600x^{49}-64x-\frac{16}{\sqrt{x}}+32\\
c) f(x) = (x^{10} – 32\sqrt{x}+32) (x^2 – 32x+32)\\
\Rightarrow f'(x)=\left [(x^{10} – 32\sqrt{x}+32) (x^2 – 32x+32) \right ]’\\
=(x^{10} – 32\sqrt{x}+32)’.(x^2 – 32x+32)+(x^{10} – 32\sqrt{x}+32) (x^2 – 32x+32)\\
=10x^9-32.\frac{1}{2\sqrt{x}}(x^2 – 32x+32)+(x^{10} – 32\sqrt{x}+32) (2x-32)\\
=10x^9-\frac{16}{\sqrt{x}}(x^2 – 32x+32)+(x^{10} – 32\sqrt{x}+32) (2x-32)\\
d) y= \frac{32x – 10 }{32x + 20}\\
\Rightarrow y’=\left (\frac{32x – 10 }{32x + 20} \right )’\\
=\frac{(32x-10)'(32x+20)-(32x-10)(32x+20)’}{(32x+20)^2}\\
=\frac{32.(32x+20)-(32x-10).32}{(32x+20)^2}\\
=\frac{1024x+640-1024x-320}{(32x+20)^2}\\
=\frac{320}{(32x+20)^2}\\
e) y= \frac{10x + 32 }{20x – 32}\\
\Rightarrow y’=\left ( \frac{10x + 32 }{20x – 32}\right )’\\
=\frac{(10x+32)'(20x-32)-(10x+32)(20x-32)’}{(20x-32)^2}\\
=\frac{10.(20x-32)-(10x+32).20}{(20x-32)^2}\\
=\frac{200x-320-200x-640}{(20x-32)^2}\\
=\frac{-960}{(20x-32)^2}\\
f) y = (32x^{20} – 32x^7 + 32x – 32)^{10}\\
\Rightarrow y’=\left [ (32x^{20} – 32x^7 + 32x – 32)^{10} \right ]’\\
=10.(32x^{20} – 32x^7 + 32x – 32)^9.(32x^{20} – 32x^7 + 32x – 32)’\\
=10.(32x^{20} – 32x^7 + 32x – 32)^9.(32.20x^{19} – 32.7x^6 + 32)\\
=10.(32x^{20} – 32x^7 + 32x – 32)^9.(640x^{19} – 224x^6 + 32)\\
g) y= sin \sqrt{32x^2 -32x +32}\\
\Rightarrow y’=\left (sin \sqrt{32x^2 -32x +32} \right )’\\
=\left (\sqrt{32x^2 -32x +32} \right )’cos \sqrt{32x^2 -32x +32}\\
=\frac{(32x^2 -32x +32)’}{2\sqrt{32x^2 -32x +32} }cos \sqrt{32x^2 -32x +32}\\
=\frac{(32.2x -32)}{2\sqrt{32x^2 -32x +32} }cos \sqrt{32x^2 -32x +32}\\
=\frac{(32x -16)}{\sqrt{32x^2 -32x +32} }cos \sqrt{32x^2 -32x +32}$