User:Angelicapulido

DERIVAR LAS SIGUIENTES FUNCIONES

1) $$\ f(x)=10x^2-9x-4$$
 * $$\ f'(x)=20x+9$$

2) $$\ f(x)=6x^3-5x^2+x-9$$
 * $$\ f'(x)=18x^2-10x+1$$

3) $$\ f(x)=(x^3-7)(2x^2+3)$$
 * $$\ f'(x)=(x^3-7)(4x)+(2x^2+3)(3x^2)$$
 * $$\ f'(x)=4x^4-28x+6x^4+9x^2$$
 * $$\ f'(x)=10x^4+9x^2-28x$$

4)$$\ f(x)=x^2(3x^4-7x+2)$$
 * $$\ f'(x)=(x^2)(12x^3-7)+(3x^4-7x+2)(2x)$$
 * $$\ f'(x)=12x^5-7x^2+6x^5-14x^2+4x$$
 * $$\ f'(x)=18x^5-21x^2+4x$$

5)$$\ f(x)=\frac {4x-5}{3x+2}$$
 * $$\ f'(x)=\frac {(3x+2)(4)-(4x-5)(3)}{(3x+2)^2}$$
 * $$\ f'(x)=\frac {12x+8-12x+15}{(3x+2)^2}$$


 * $$\ f'(x)=\frac {23}{(3x+2)^2}$$

6)$$\ f(x)=\frac {8-x+3x^2}{2-9x}$$
 * $$\ f'(x)=\frac {(2-9x)(6x-1)-(8-x+3x^2)(-9)}{(2-9x)^2}$$
 * $$\ f'(x)=\frac {12x-2-54x^2+9x+72-9x+27x^2}{(2-9x)^2}$$
 * $$\ f'(x)=\frac {12x-27x^2+70}{(2-9x)^2}$$

7)$$\ f(x)=\frac {1}{1+x+x^2+x^3}$$
 * $$\ f'(x)=\frac {(1+x+x^2+x^3)(0)-(1)(1+2x+3x^2)}{(1+x+x^2+x^3)^2}$$
 * $$\ f'(x)=\frac {0-1+2x+3x^2}{(1+x+x^2+x^3)^2}$$
 * $$\ f'(x)=\frac {-3x^2-2x-1}{(1+x+x^2+x^3)^2}$$

8)$$\ f(x)=\frac {3x-1}{x^2}$$
 * $$\ f'(x)=\frac {(x^2)(3)-(3x-1)(2x)}{(x^2)^2}$$
 * $$\ f'(x)=\frac {3x^2-6x^2-2x}{(x^4}$$
 * $$\ f'(x)=\frac {-3x^2-2x}{(x^4)}$$

9)$$\ x^2+y^2=1$$
 * $$\ 2x+2y\frac {dy}{dx}=0$$
 * $$\ 2y\frac {dy}{dx}=-2x$$
 * $$\ \frac {dy}{dx}=\frac {-2x}{2y}$$
 * $$\ \frac {dy}{dx}=\frac {-x}{y}$$

10)$$\ y^2=\frac {x-1}{x+1}$$
 * $$\ 2y\frac {dy}{dx}=\frac {(x+1)-(x-1)}{(x+1)^2}$$
 * $$\ 2y\frac {dy}{dx}=\frac {x+1-x+1}{(x+1)^2}$$
 * $$\ \frac {dy}{dx}=\frac {2}{2y(x+1)^2}$$
 * $$\ \frac {dy}{dx}=\frac {1}{y(x+1)^2}$$

11)$$\ x^2+xy=2$$
 * $$\ 2x+x\frac {dy}{dx}+y=0$$
 * $$\ x\frac {dy}{dx}=-2x-y$$
 * $$\ \frac {dy}{dx}=\frac {-2x+y}{x}$$

12)$$\ x^2y+xy^2=6$$
 * $$\ x^2\frac {dy}{dx}+2xy+2xy\frac {dy}{dx}+y^2=0$$
 * $$\ x^2\frac {dy}{dx}+2xy\frac {dy}{dx}=-2xy-y^2$$
 * $$\ \frac {dy}{dx}(x^2+2xy)=-2xy-y^2$$
 * $$\ \frac {dy}{dx}=\frac {-2xy-y^2}{x^2+2xy}$$

13)$$\ \frac {1}{y}+\frac {1}{x}=1$$
 * $$\ y^-1+x^-1=0$$
 * $$\ -y^-2\frac {dy}{dx}-x^-2=0$$
 * $$\ -y^-2\frac {dy}{dx}=x^-2$$
 * $$\ \frac {dy}{dx}=\frac {x^-2}{-y^-2}$$

14)$$\ y^2=x^2(x^2+1)$$
 * $$\ 2y\frac {dy}{dx}=2x(x^2+1)+(2x)(x^2)$$
 * $$\ 2y\frac {dy}{dx}=2x(x^2+1+x^2)$$
 * $$\ \frac {dy}{dx}=\frac {2x(x^2+1+x^2)}{2y}$$
 * $$\ \frac {dy}{dx}=\frac {x(2x^2+1)}{y}$$

15)$$\ x^2y^2=x^2+y^2$$
 * $$\ 2yx^2\frac {dy}{dx}+2xy^2=2x+2y\frac {dy}{dx}$$
 * $$\ 2yx^2\frac {dy}{dx}-2y\frac {dy}{dx}=2x-2xy^2$$
 * $$\ 2y\frac {dy}{dx}(x^2-1)=2x-2xy^2$$
 * $$\ \frac {dy}{dx}= \frac {2x-2xy^2}{2y(x^2-1)}$$
 * $$\ \frac {dy}{dx}= \frac {2x(1-y^2)}{2y(x^2-1)}$$
 * $$\ \frac {dy}{dx}= \frac {x(1-y^2)}{y(x^2-1)}$$

HALLAR EL LIMITE DE LAS SIGUIENTES FUNCIONES

dada :$$\ f(x)=x^2-3x$$ hallar:$$\lim_{h\to 0}\frac {f(x+h)-f(x)}{h}$$
 * $$\lim_{h\to 0}\frac {(x+h)^2-3(x+h)-(x^2-3x)}{h}$$
 * $$\lim_{h\to 0}\frac {x^2+2xh+h^2-3x-3h-x^2+3x}{h}$$
 * $$\lim_{h\to 0}\frac {2xh+h^2-3h}{h}$$
 * $$\lim_{h\to 0}2x+h-3=2x-3$$