Chapter 14 - A Preview of Calculus: The Limit, Derivative, and Integral of a Function - 14.4 The Tangent Problem; The Derivative - 14.4 Assess Your Understanding - Page 898: 21

$-4$

The derivative of a function $f(x)$ at $a$ is: $\lim_{x\to a}\frac{f(x)-f(a)}{x-a}$. We do what we have to according to the exercise: $$f'(3)=\lim_{x\to 3}\frac{f(x)-f(3)}{x-3}=\lim_{x\to 3}\frac{-4x+5-(-4(3)+5)}{x-3}=\lim_{x\to 3}\frac{-4x+5-(-7)}{x-3}=\lim_{x\to 3}\frac{-4x+12}{x-3}=\lim_{x\to 3}\frac{-4(x-3)}{x-3}=\lim_{x\to 3}-4=-4.$$