Find \(f'(x)\) from first principles

\(f(x)=-7x^{2}-2x-6\)

\begin{align} f'(x) & = \lim_{\delta x\to \; 0} \quad \frac{f(x+\delta x)-f(x)}{\delta x}\\ & = \lim_{\delta x\to \; 0} \quad \frac{-7(x+\delta x)^2-2(x+ \delta x)-6 -(-7x^{2}-2x-6)}{\delta x} \\ & = \lim_{\delta x\to \; 0} \quad \frac{-7(x^2+2x\delta x +(\delta x)^2) )-2x -2 \delta x -6 +7x^2+2x+6}{\delta x}\\ & = \lim_{\delta x\to \; 0} \quad \frac{-7x^2-14x\delta x -7(\delta x)^2 -2x-2\delta x -6 +7x^2 +2x +6}{\delta x}\\ & = \lim_{\delta x\to \; 0} \quad \frac {-14x \delta x -7(\delta x)^2 -2 \delta x }{\delta x} \\ & = \lim_{\delta x\to \; 0} \quad -14x-7 \delta x -2 \\ & = \quad \quad \quad -14x -2 \end{align}