[tex]\displaystyle
f'(x)=\lim_{\Delta x\to 0}\dfrac{-3(x+\Delta x)^2-9(x+\Delta x)-(-3x^2-9x)}{\Delta x}\\
f'(x)=\lim_{\Delta x\to 0}\dfrac{-3x^2-6x\Delta x-3(\Delta x)^2-9x-9\Delta x+3x^2+9x}{\Delta x}\\
f'(x)=\lim_{\Delta x\to 0}\dfrac{-6x\Delta x-3(\Delta x)^2-9\Delta x}{\Delta x}\\
f'(x)=\lim_{\Delta x\to 0}\dfrac{\Delta x(-6x-3x\Delta x-9)}{\Delta x}\\
f'(x)=\lim_{\Delta x\to 0}(-6x-3x\Delta x-9)\\
f'(x)=-6x-3x\cdot 0-9\\
f'(x)=-6x-9\Rightarrow \text{C}
[/tex]
