¿Cuál es la derivada de la siguiente función? \[ f(x)=-5 x^{3} \] a. \( \frac{15}{x^{(-2)}} \) b. \( \frac{-15 x^{2}}{x^{4}} \) d. \( \frac{-15}{x^{-2}} \)

11. Find the values of \( a \) and \( b \) that make \( f \) continuous everywhere. \[ f(x)=\left\{\begin{array}{ll}\frac{x^{2}-4}{x-2} & \text { if } x<2 \\ a x^{2}-b x+3 & \text { if } 2 \leq x<3 \\ 2 x-a+b & \text { if } x \geq 3\end{array}\right. \]

Show that \( F(x)=|x| \) is continuous everywhere.

Calcule \( \int \frac{x^{4} - 5x^{2} + 6}{x-2} \,dx \) aplicando o método de divisão longa.

USing first principle prove that \( y=\sin x \) then \( \frac{d y}{d x}=\cos x \)

7. Find \( \lim _{x \rightarrow 2} \frac{\sqrt{6-x}-2}{\sqrt{3-x}-1} \) 8. If \( f(x)=x^{2}+10 \sin x \), show that there is a number \( c \) such that \( f(x)=1000 \)