1.
a) 2x²-6x-3x²-15x=-x<span>²-21x
б) a</span>²+7a-a-7-(a²-6a+9)=a²+6a-7-a<span>²+6a-9=12a-16
в) 3(y</span>²+10y+25)-3y²=3y²+30y+75-3y<span>²=30y+75
2.
a) c(c</span>²-16)=c(c-4)(c+4)
б) 3(a²-2ab+b²)=3(a-b)<span>²
в) (4-x</span>²)(4+x²)=(2-x)(2+x)(4+x²)
3.
9a<span>²-6a</span>³+a⁴-a²(a²-4)+14a+6a³=9a²-6a³+a⁴-a⁴+4a²+14a+6a³=13a²+14a
4.
a) x⁴+3x²-x²-3=1+2x²+x⁴+x
x⁴-x⁴+2x²-2x²-x=1+3
-x=4
x= -4
Ответ: -4
б) 4x²-20x+25-4x²+9=0
-20x=-34
x=1.7
Ответ: 1,7.
5.
a) (9a²-4)(9a²+4)=(3a-2)(3a+2)(9a²+4)
б) (a-b)+(a²-b²)=(a-b)+(a-b)(a+b)=(a-b)(1+a+b)
в) x²-(y²-4y+4)=x²-(y-2)²=(x-y+2)(x+y-2)
6.
-x²-12x-41= -(x²+12x+41)= - (x² +12x+36+5)=
= - ((x+6)² + 5) = -5 - (x+6)² <0 при любом значении х.
Правило дифференцирования сложной функции:
(u(v))' = u'(v)*v'
1)
![f(x)=(x^3-x^5+6)^7=[v=x^3-x^5+6; u=v^7]=(v^7)'*v'=7v^{7-1}*(x^3-x^5+6)'=7v^6*(3x^{3-1}-5x^{5-1})=7(x^3-x^5+6)^6*(3x^{2}-5x^{4})](https://tex.z-dn.net/?f=f%28x%29%3D%28x%5E3-x%5E5%2B6%29%5E7%3D%5Bv%3Dx%5E3-x%5E5%2B6%3B+u%3Dv%5E7%5D%3D%28v%5E7%29%27%2Av%27%3D7v%5E%7B7-1%7D%2A%28x%5E3-x%5E5%2B6%29%27%3D7v%5E6%2A%283x%5E%7B3-1%7D-5x%5E%7B5-1%7D%29%3D7%28x%5E3-x%5E5%2B6%29%5E6%2A%283x%5E%7B2%7D-5x%5E%7B4%7D%29)
2)
![f(x)=((5-3x^3+x^4)^6)=[v=5-3x^3+x^4; u=v^6]=(v^6)'*v'=6v^{6-1}*(5-3x^3+x^4)'=6v^5*(-3*3x^{3-1}+4x^{4-1})=6(5-3x^3+x^4)^5*(-9x^2+4x^3)](https://tex.z-dn.net/?f=f%28x%29%3D%28%285-3x%5E3%2Bx%5E4%29%5E6%29%3D%5Bv%3D5-3x%5E3%2Bx%5E4%3B+u%3Dv%5E6%5D%3D%28v%5E6%29%27%2Av%27%3D6v%5E%7B6-1%7D%2A%285-3x%5E3%2Bx%5E4%29%27%3D6v%5E5%2A%28-3%2A3x%5E%7B3-1%7D%2B4x%5E%7B4-1%7D%29%3D6%285-3x%5E3%2Bx%5E4%29%5E5%2A%28-9x%5E2%2B4x%5E3%29)
3)
![f(x)=(3\sqrt{x}+2x)^5=5(3\sqrt{x}+2x)^{5-1}*(3\sqrt{x}+2x)'=\\\\=5(3\sqrt{x}+2x)^{4}*(3x^{\frac{1}{2}}+2x)'=5(3\sqrt{x}+2x)^{4}*(3*\frac{1}{2}x^{\frac{1}{2}-1}+2x^{1-1})=5(3\sqrt{x}+2x)^{4}*(\frac{3}{2}x^{-\frac{1}{2}}+2x^{1-1})=5(3\sqrt{x}+2x)^{4}*(\frac{3}{2\sqrt{x}}+2x^{0})=5(3\sqrt{x}+2x)^{4}*(\frac{3}{2\sqrt{x}}+2)](https://tex.z-dn.net/?f=f%28x%29%3D%283%5Csqrt%7Bx%7D%2B2x%29%5E5%3D5%283%5Csqrt%7Bx%7D%2B2x%29%5E%7B5-1%7D%2A%283%5Csqrt%7Bx%7D%2B2x%29%27%3D%5C%5C%5C%5C%3D5%283%5Csqrt%7Bx%7D%2B2x%29%5E%7B4%7D%2A%283x%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D%2B2x%29%27%3D5%283%5Csqrt%7Bx%7D%2B2x%29%5E%7B4%7D%2A%283%2A%5Cfrac%7B1%7D%7B2%7Dx%5E%7B%5Cfrac%7B1%7D%7B2%7D-1%7D%2B2x%5E%7B1-1%7D%29%3D5%283%5Csqrt%7Bx%7D%2B2x%29%5E%7B4%7D%2A%28%5Cfrac%7B3%7D%7B2%7Dx%5E%7B-%5Cfrac%7B1%7D%7B2%7D%7D%2B2x%5E%7B1-1%7D%29%3D5%283%5Csqrt%7Bx%7D%2B2x%29%5E%7B4%7D%2A%28%5Cfrac%7B3%7D%7B2%5Csqrt%7Bx%7D%7D%2B2x%5E%7B0%7D%29%3D5%283%5Csqrt%7Bx%7D%2B2x%29%5E%7B4%7D%2A%28%5Cfrac%7B3%7D%7B2%5Csqrt%7Bx%7D%7D%2B2%29)
Sin(2x)+sin(22x)=cos(23x)+cos(24x)