Use definition (2) (p. 135) to find the slope of the line tangent to the graph of f at P.
f(x) = x³; 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 \( f(x) = x^3 \) and the point P is (1, 1). We need to find \( f'(1) \).
3. Step 3: Substitute the function into the derivative definition. Calculate \( f(1+h) = (1+h)^3 \) and \( f(1) = 1^3 = 1 \).
4. Step 4: Expand \( (1+h)^3 \) using the binomial theorem: \( (1+h)^3 = 1 + 3h + 3h^2 + h^3 \).
5. Step 5: Substitute these into the derivative formula: \( f'(1) = \lim_{h \to 0} \frac{(1 + 3h + 3h^2 + h^3) - 1}{h} \). Simplify the expression and evaluate the limit as \( h \to 0 \).

Key Concepts

Tangent Line

A tangent line to a curve at a given point is a straight line that touches the curve at that point without crossing it. The slope of the tangent line represents the instantaneous rate of change of the function at that point, which is crucial for understanding how the function behaves locally.

Derivative

The derivative of a function at a point quantifies how the function's output changes as its input changes. It is defined as the limit of the average rate of change of the function as the interval approaches zero. For the function f(x) = x^3, the derivative can be calculated using the power rule.

Power Rule

The power rule is a basic differentiation rule that states if f(x) = x^n, then f'(x) = n*x^(n-1). This rule simplifies the process of finding derivatives for polynomial functions, making it essential for calculating the slope of the tangent line for functions like f(x) = x^3.