Question

calculo diferencial
y= 10x2+1
y= 8x2+3

Answer 1

1)
Por propiedades:
${y=10x^2+1}\\\\\\{\dfrac{dy}{dx}=\dfrac{d}{dx}(10x^2)+\dfrac{d}{dx}(1)}\\\\\\{y'=10\frac{d}{dx}(x^2)+0}\\\\{y'=10(2x) \to 20x}$Por demostración formal:
$y'= \lim_{\triangle x \to 0} \dfrac{f(x+\triangle x)-f(x)}{\triangle x}}\\\\\\{y'= \lim_{\triangle x \to 0} \dfrac{10(x+\triangle x)^2+1-10x^2-1}{\triangle x}}\\\\\\{y'= \lim_{\triangle x \to 0} \dfrac{10x^2+20x. \triangle x +10 \triangle x ^2+1-10x^2-1}{\triangle x}}\\\\\\{y'= \lim_{\triangle x\to 0}\dfrac{20x.\triangle x+10\triangle x ^2}{\triangle x}}\\\\\\{y'= \lim_{\triangle x \to 0} \dfrac{10\triangle x(2x+\triangle x)}{\triangle x}}\\\\\\{y'= \lim_{\triangle x \to 0}10(2x+\triangle x)}\\\\\\{ y'=10(2x+0)\to\boxed{ y'=20x}$