Question

$$\frac{x^2-1}{x}$$

Answer 1

$$\frac{d}{dx}(f(x))=\underset{h\to 0}{lim}\frac{f(x+h)-f\left(x\right)}{h}$$
$$=\underset{h\to 0}{lim}\frac{\frac{(x+h{)}^2-1}{x+h}-\frac{x^2-1}{x}}{h}$$
$$=\underset{h\to 0}{lim}\frac{\frac{x^2+2xh+h^2-1}{x+h}-\frac{x^2-1}{x}}{h}$$
$$=\underset{h\to 0}{lim}\frac{x^3+2x^{2}h+h^{2}x-x-x^3-x^{2}h+x+h}{xh(x+h)}$$
$$=\underset{h\to 0}{lim}\frac{x^{2}h+h^{2}x+h}{x(x+h)}$$
$$=\underset{h\to 0}{lim}\frac{h(x^2+hx+1)}{xh(x+h)}$$
$$=\underset{h\to 0}{lim}\frac{x^2+hx+1}{x(x+h)}$$
$$=\frac{x^2+1}{x^2}$$