Ответы
Ответ дал:
1
Не во всех примерах я доводила до конца преобразования
Приложения:
![](https://st.uroker.com/files/7c6/7c69c398a4ac32c5117e4cdc712adf6f.jpg)
![](https://st.uroker.com/files/e2c/e2cf883a6f09c345456382633dc5e41d.jpg)
m11m:
В 5 номере производная (х^3)=3x^2, а у вас 2х^2.
Ответ дал:
0
1.
![f'(x)=2x- \frac{1}{ x^{2} }= \frac{2x^3-1}{ x^{2} } f'(x)=2x- \frac{1}{ x^{2} }= \frac{2x^3-1}{ x^{2} }](https://tex.z-dn.net/?f=f%27%28x%29%3D2x-+%5Cfrac%7B1%7D%7B+x%5E%7B2%7D+%7D%3D+%5Cfrac%7B2x%5E3-1%7D%7B+x%5E%7B2%7D+%7D++)
2.
![f'(x)=-5x^4 f'(x)=-5x^4](https://tex.z-dn.net/?f=f%27%28x%29%3D-5x%5E4)
3.
![f'(x)=-6x^2+ \frac{6}{ \sqrt{x} } f'(x)=-6x^2+ \frac{6}{ \sqrt{x} }](https://tex.z-dn.net/?f=f%27%28x%29%3D-6x%5E2%2B+%5Cfrac%7B6%7D%7B+%5Csqrt%7Bx%7D+%7D+)
4.
![f(x)= \frac{7}{ \sqrt[4]{x} }- \frac{3}{x}=7*x^{- \frac{1}{4} }- \frac{3}{x} \\ \\
f'(x)=- \frac{7}{4x \sqrt{x} } + \frac{3}{x^2}= \frac{12-7 \sqrt{x} }{4x^2} f(x)= \frac{7}{ \sqrt[4]{x} }- \frac{3}{x}=7*x^{- \frac{1}{4} }- \frac{3}{x} \\ \\
f'(x)=- \frac{7}{4x \sqrt{x} } + \frac{3}{x^2}= \frac{12-7 \sqrt{x} }{4x^2}](https://tex.z-dn.net/?f=f%28x%29%3D+%5Cfrac%7B7%7D%7B+%5Csqrt%5B4%5D%7Bx%7D+%7D-+%5Cfrac%7B3%7D%7Bx%7D%3D7%2Ax%5E%7B-+%5Cfrac%7B1%7D%7B4%7D+%7D-+%5Cfrac%7B3%7D%7Bx%7D+%5C%5C++%5C%5C+%0Af%27%28x%29%3D-+%5Cfrac%7B7%7D%7B4x+%5Csqrt%7Bx%7D+%7D++++%2B+%5Cfrac%7B3%7D%7Bx%5E2%7D%3D+%5Cfrac%7B12-7+%5Csqrt%7Bx%7D+%7D%7B4x%5E2%7D++)
5.
![f(x)=(x-6)x^3=x^4-6x^3 \\ \\
f'(x)=4x^3-18x^2 f(x)=(x-6)x^3=x^4-6x^3 \\ \\
f'(x)=4x^3-18x^2](https://tex.z-dn.net/?f=f%28x%29%3D%28x-6%29x%5E3%3Dx%5E4-6x%5E3+%5C%5C++%5C%5C+%0Af%27%28x%29%3D4x%5E3-18x%5E2)
6.
![f(x)= \sqrt{x} (x^2+ \frac{1}{ \sqrt{x} } )=x^{2.5}+1 \\ \\
f'(x)=2.5x^{1.5}=2.5x \sqrt{x} f(x)= \sqrt{x} (x^2+ \frac{1}{ \sqrt{x} } )=x^{2.5}+1 \\ \\
f'(x)=2.5x^{1.5}=2.5x \sqrt{x}](https://tex.z-dn.net/?f=f%28x%29%3D+%5Csqrt%7Bx%7D+%28x%5E2%2B+%5Cfrac%7B1%7D%7B+%5Csqrt%7Bx%7D+%7D+%29%3Dx%5E%7B2.5%7D%2B1+%5C%5C++%5C%5C+%0Af%27%28x%29%3D2.5x%5E%7B1.5%7D%3D2.5x+%5Csqrt%7Bx%7D+)
7.
![f'(x)= \frac{6}{2 \sqrt{6x+1} }*(x^4-5)+4x^3 \sqrt{6x+1}= \\ \\
= \frac{3(x^4-5)}{ \sqrt{6x+1} }+4x^3 \sqrt{6x+1}= \frac{3x^4-15+4x^3(6x+1)}{ \sqrt{6x+1} }= \\ \\
= \frac{3x^4-15+24x^4+4x^3}{ \sqrt{6x+1} }= \frac{27x^4+4x^3-15}{ \sqrt{6x+1} } f'(x)= \frac{6}{2 \sqrt{6x+1} }*(x^4-5)+4x^3 \sqrt{6x+1}= \\ \\
= \frac{3(x^4-5)}{ \sqrt{6x+1} }+4x^3 \sqrt{6x+1}= \frac{3x^4-15+4x^3(6x+1)}{ \sqrt{6x+1} }= \\ \\
= \frac{3x^4-15+24x^4+4x^3}{ \sqrt{6x+1} }= \frac{27x^4+4x^3-15}{ \sqrt{6x+1} }](https://tex.z-dn.net/?f=f%27%28x%29%3D+%5Cfrac%7B6%7D%7B2+%5Csqrt%7B6x%2B1%7D+%7D%2A%28x%5E4-5%29%2B4x%5E3+%5Csqrt%7B6x%2B1%7D%3D+%5C%5C++%5C%5C+%0A%3D+%5Cfrac%7B3%28x%5E4-5%29%7D%7B+%5Csqrt%7B6x%2B1%7D+%7D%2B4x%5E3+%5Csqrt%7B6x%2B1%7D%3D+%5Cfrac%7B3x%5E4-15%2B4x%5E3%286x%2B1%29%7D%7B+%5Csqrt%7B6x%2B1%7D+%7D%3D+%5C%5C++%5C%5C+%0A%3D+%5Cfrac%7B3x%5E4-15%2B24x%5E4%2B4x%5E3%7D%7B+%5Csqrt%7B6x%2B1%7D+%7D%3D+%5Cfrac%7B27x%5E4%2B4x%5E3-15%7D%7B+%5Csqrt%7B6x%2B1%7D+%7D+++++++)
8.
![f'(x)=( \frac{x}{3}+1 )^3+x*3( \frac{x}{3}+1 )^2* \frac{1}{3}= \\ \\
=( \frac{x}{3}+1 )^3+x( \frac{x}{3}+1 )^2=( \frac{x}{3}+1 )^2( \frac{x}{3}+1+x )= \\ \\
=( \frac{x}{3}+1 )^2( \frac{x+3+3x}{3} )= \frac{1}{3}(4x+3)( \frac{x}{3}+1 )^2 f'(x)=( \frac{x}{3}+1 )^3+x*3( \frac{x}{3}+1 )^2* \frac{1}{3}= \\ \\
=( \frac{x}{3}+1 )^3+x( \frac{x}{3}+1 )^2=( \frac{x}{3}+1 )^2( \frac{x}{3}+1+x )= \\ \\
=( \frac{x}{3}+1 )^2( \frac{x+3+3x}{3} )= \frac{1}{3}(4x+3)( \frac{x}{3}+1 )^2](https://tex.z-dn.net/?f=f%27%28x%29%3D%28+%5Cfrac%7Bx%7D%7B3%7D%2B1+%29%5E3%2Bx%2A3%28+%5Cfrac%7Bx%7D%7B3%7D%2B1+%29%5E2%2A+%5Cfrac%7B1%7D%7B3%7D%3D+%5C%5C++%5C%5C+%0A%3D%28+%5Cfrac%7Bx%7D%7B3%7D%2B1+%29%5E3%2Bx%28+%5Cfrac%7Bx%7D%7B3%7D%2B1+%29%5E2%3D%28+%5Cfrac%7Bx%7D%7B3%7D%2B1+%29%5E2%28+%5Cfrac%7Bx%7D%7B3%7D%2B1%2Bx+%29%3D+%5C%5C++%5C%5C+%0A%3D%28+%5Cfrac%7Bx%7D%7B3%7D%2B1+%29%5E2%28+%5Cfrac%7Bx%2B3%2B3x%7D%7B3%7D+%29%3D+%5Cfrac%7B1%7D%7B3%7D%284x%2B3%29%28+%5Cfrac%7Bx%7D%7B3%7D%2B1+%29%5E2++)
9.
![f'(x)= \frac{2(3-2x)+2(2x+3)}{(3-2x)^2}= \frac{6-4x+4x+6}{(3-2x)^2}= \frac{12}{(3-2x)^2} f'(x)= \frac{2(3-2x)+2(2x+3)}{(3-2x)^2}= \frac{6-4x+4x+6}{(3-2x)^2}= \frac{12}{(3-2x)^2}](https://tex.z-dn.net/?f=f%27%28x%29%3D+%5Cfrac%7B2%283-2x%29%2B2%282x%2B3%29%7D%7B%283-2x%29%5E2%7D%3D+%5Cfrac%7B6-4x%2B4x%2B6%7D%7B%283-2x%29%5E2%7D%3D+%5Cfrac%7B12%7D%7B%283-2x%29%5E2%7D+++)
10.
![f'(x)= \frac{3x^2(2x-3)-2x^3}{(2x-3)^2}= \frac{6x^3-9x^2-2x^3}{(2x-3)^2}= \frac{4x^3-9x^2}{(2x-3)^2} f'(x)= \frac{3x^2(2x-3)-2x^3}{(2x-3)^2}= \frac{6x^3-9x^2-2x^3}{(2x-3)^2}= \frac{4x^3-9x^2}{(2x-3)^2}](https://tex.z-dn.net/?f=f%27%28x%29%3D+%5Cfrac%7B3x%5E2%282x-3%29-2x%5E3%7D%7B%282x-3%29%5E2%7D%3D+%5Cfrac%7B6x%5E3-9x%5E2-2x%5E3%7D%7B%282x-3%29%5E2%7D%3D+%5Cfrac%7B4x%5E3-9x%5E2%7D%7B%282x-3%29%5E2%7D+++)
11.
![f'(x)= \frac{(4x^3+2x)(x+1)-(x^4+x^2+1)}{(x+1)^2}= \\ \\
= \frac{4x^4+2x^2+4x^3+2x-x^4-x^2-x}{(x+1)^2}= \\ \\
= \frac{3x^4+4x^3+x^2+2x-1}{(x+1)^2} f'(x)= \frac{(4x^3+2x)(x+1)-(x^4+x^2+1)}{(x+1)^2}= \\ \\
= \frac{4x^4+2x^2+4x^3+2x-x^4-x^2-x}{(x+1)^2}= \\ \\
= \frac{3x^4+4x^3+x^2+2x-1}{(x+1)^2}](https://tex.z-dn.net/?f=f%27%28x%29%3D+%5Cfrac%7B%284x%5E3%2B2x%29%28x%2B1%29-%28x%5E4%2Bx%5E2%2B1%29%7D%7B%28x%2B1%29%5E2%7D%3D+%5C%5C++%5C%5C+%0A%3D+%5Cfrac%7B4x%5E4%2B2x%5E2%2B4x%5E3%2B2x-x%5E4-x%5E2-x%7D%7B%28x%2B1%29%5E2%7D%3D+%5C%5C++%5C%5C+%0A%3D+%5Cfrac%7B3x%5E4%2B4x%5E3%2Bx%5E2%2B2x-1%7D%7B%28x%2B1%29%5E2%7D+++)
12.
![f'(x)= \frac{2x^5(3x-2)-3( \frac{1}{3}x^6+2 )}{(3x-2)^2}= \frac{6x^6-4x^5-x^6-6}{(3x-2)^2}= \frac{5x^6-4x^5-6}{(3x-2)^2} f'(x)= \frac{2x^5(3x-2)-3( \frac{1}{3}x^6+2 )}{(3x-2)^2}= \frac{6x^6-4x^5-x^6-6}{(3x-2)^2}= \frac{5x^6-4x^5-6}{(3x-2)^2}](https://tex.z-dn.net/?f=f%27%28x%29%3D+%5Cfrac%7B2x%5E5%283x-2%29-3%28+%5Cfrac%7B1%7D%7B3%7Dx%5E6%2B2+%29%7D%7B%283x-2%29%5E2%7D%3D+%5Cfrac%7B6x%5E6-4x%5E5-x%5E6-6%7D%7B%283x-2%29%5E2%7D%3D+%5Cfrac%7B5x%5E6-4x%5E5-6%7D%7B%283x-2%29%5E2%7D+++)
13.
![f'(x)= \frac{15x^2(x-4)^2-5x^3*2(x-4)}{(x-4)^4}= \frac{15x^2(x^2-8x+16)-10x^4+40x^3}{(x-4)^4}= \\ \\
= \frac{15x^4-120x^3+240x^2-10x^4+40x^3}{(x-4)^4}= \frac{5x^4-80x^3+240x^2}{(x-4)^4} f'(x)= \frac{15x^2(x-4)^2-5x^3*2(x-4)}{(x-4)^4}= \frac{15x^2(x^2-8x+16)-10x^4+40x^3}{(x-4)^4}= \\ \\
= \frac{15x^4-120x^3+240x^2-10x^4+40x^3}{(x-4)^4}= \frac{5x^4-80x^3+240x^2}{(x-4)^4}](https://tex.z-dn.net/?f=f%27%28x%29%3D+%5Cfrac%7B15x%5E2%28x-4%29%5E2-5x%5E3%2A2%28x-4%29%7D%7B%28x-4%29%5E4%7D%3D+%5Cfrac%7B15x%5E2%28x%5E2-8x%2B16%29-10x%5E4%2B40x%5E3%7D%7B%28x-4%29%5E4%7D%3D+%5C%5C++%5C%5C+%0A%3D+%5Cfrac%7B15x%5E4-120x%5E3%2B240x%5E2-10x%5E4%2B40x%5E3%7D%7B%28x-4%29%5E4%7D%3D+%5Cfrac%7B5x%5E4-80x%5E3%2B240x%5E2%7D%7B%28x-4%29%5E4%7D++++)
14.
![f'(x)= \frac{2x(x^3-x)+(x^2+1)(3x^2-1)}{(x^3-x)^2}= \frac{2x^4-2x^2+3x^4+3x^2-x^2-1}{(x^3-x)^2}= \\ \\
= \frac{5x^4-1}{(x^3-x)^2} f'(x)= \frac{2x(x^3-x)+(x^2+1)(3x^2-1)}{(x^3-x)^2}= \frac{2x^4-2x^2+3x^4+3x^2-x^2-1}{(x^3-x)^2}= \\ \\
= \frac{5x^4-1}{(x^3-x)^2}](https://tex.z-dn.net/?f=f%27%28x%29%3D+%5Cfrac%7B2x%28x%5E3-x%29%2B%28x%5E2%2B1%29%283x%5E2-1%29%7D%7B%28x%5E3-x%29%5E2%7D%3D+%5Cfrac%7B2x%5E4-2x%5E2%2B3x%5E4%2B3x%5E2-x%5E2-1%7D%7B%28x%5E3-x%29%5E2%7D%3D+%5C%5C++%5C%5C+%0A%3D+%5Cfrac%7B5x%5E4-1%7D%7B%28x%5E3-x%29%5E2%7D+++)
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