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

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 at a point x = a is given by the limit: \( f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h} \).
2. Step 2: Identify the function and the point of interest. Here, the function is \( f(x) = \frac{1}{x} \) and the point P is (1, 1). We need to find \( f'(1) \).
3. Step 3: Substitute \( f(x) = \frac{1}{x} \) into the derivative formula. This gives: \( f'(1) = \lim_{h \to 0} \frac{f(1+h) - f(1)}{h} = \lim_{h \to 0} \frac{\frac{1}{1+h} - 1}{h} \).
4. Step 4: Simplify the expression inside the limit. Start by finding a common denominator for the terms in the numerator: \( \frac{1}{1+h} - 1 = \frac{1 - (1+h)}{1+h} = \frac{-h}{1+h} \).
5. Step 5: Substitute the simplified expression back into the limit: \( f'(1) = \lim_{h \to 0} \frac{-h}{h(1+h)} \). Simplify this to \( \lim_{h \to 0} \frac{-1}{1+h} \) and evaluate the limit as \( h \to 0 \).