Question

Find the derivative of the following function using the limit process

Answer 1

`\displaystyle f'(x)=\lim_(\Delta x\to 0)(-3(x+\Delta x)^2-9(x+\Delta x)-(-3x^2-9x))/(\Delta x)\\ f'(x)=\lim_(\Delta x\to 0)(-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)(-6x\Delta x-3(\Delta x)^2-9\Delta x)/(\Delta x)\\ f'(x)=\lim_(\Delta x\to 0)(\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}`