Utilize a definição de derivadas encontre a derivada de f(x) = x²

f'(x)=2x

Seja \(f(x)\) uma função diferenciável num intervalo \(I\), definimos nesse intervalo, a derivada de \(f\) como o limite:

                                                                         \(f'(x)=\lim_{h \to 0} \frac{f(x+h)-f(x)}{h} \)

Seja \(f(x)=x^2\) aplicando a definição de derivada, temos:

                                                                         \(f'(x)=\lim_{h \to 0} \frac{(x+h)^2-x^2}{h} \)

                                                                                   \(=\lim_{h \to 0} \frac{x^2+2hx+h^2-x^2}{h} \)

                                                                                   \(=\lim_{h \to 0} \frac{2hx+h^2}{h} \)

                                                                                   \(=\lim_{h \to 0} 2x+h\)

                                                                                   \(=2x\)

Portanto, temos que, pela definição de derivada:

                                                                          \(f'(x)=2x\)