- Inicio
- Derivatives
- Derivative at a point
- Ejercicios
Derivative at a point
Given the function $f(x)=x^2+x$, compute the derivative in the point $x=1$.
Using the definition $$\displaystyle f'(a)=\lim_{\Delta x \to 0}\frac{f(a+\Delta x)-f(a)}{\Delta x}$$ At the point $x=1$
$$\displaystyle f'(1)=\lim_{\Delta x \to 0}\frac{f(1+\Delta x)-f(1)}{\Delta x}=\lim_{\Delta x \to 0}\frac{(1+\Delta x)^2+(1+\Delta x)-(1^2+1)}{\Delta x}=$$ $$=\lim_{\Delta x \to 0}\dfrac{\Delta x^2+3\Delta x}{\Delta x}=\lim_{\Delta x \to 0}(\Delta x+3)=3 $$
$$f'(1)=3$$