Question

Let P and Q be points on the curve 𝑦=𝑥2−2𝑥 while x=2 and x=2+h respectively. Express the gradient of pq in terms of h and hence find the gradient of the curve 𝑦=𝑥2−2𝑥 at x= 2

Answer 1

$$\dfrac{(x+h)^2-2(x+h)-(x^2-2x)}{x+h-x}$$

$$=\dfrac{x^2+2xh+h^2-2x-2h-x^2+2x}{h}$$

$$=2x-2+h$$

$$y'(x)=\lim\limits_{h\to 0}\dfrac{(x+h)^2-2(x+h)-(x^2-2x)}{x+h-x}$$

$$=\lim\limits_{h\to 0}(2x-2+h)=2x-2+0=2x-2$$

$$\text{grad}\ y=y'=2x-2$$

$x=2$

$$\text{grad}\ y|_{x=2}=y'(2)=2(2)-2=2$$

$$\text{grad}\ y|_{x=2}=2$$