Use definition (2) (p. 135) to find the slope of the line tangent to the graph of f at P.
f(x) = -7x; P(-1,7)

Verified step by step guidance

1. Step 1: Recall the definition of the derivative as the slope of the tangent line at a point. The derivative of a function \( f(x) \) at a point \( x = a \) is given by \( f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h} \).
2. Step 2: Identify the function \( f(x) = -7x \) and the point \( P(-1, 7) \). Here, \( a = -1 \) and \( f(a) = 7 \).
3. Step 3: Substitute \( f(x) = -7x \) into the derivative definition: \( f'(a) = \lim_{h \to 0} \frac{f(-1+h) - f(-1)}{h} \).
4. Step 4: Calculate \( f(-1+h) \) and \( f(-1) \). Since \( f(x) = -7x \), we have \( f(-1+h) = -7(-1+h) = 7 - 7h \) and \( f(-1) = -7(-1) = 7 \).
5. Step 5: Substitute these values into the limit expression: \( f'(a) = \lim_{h \to 0} \frac{(7 - 7h) - 7}{h} \). Simplify the expression and evaluate the limit to find the slope of the tangent line.