Use definition (1) (p. ​133​) to find the slope of the line tangent to the graph of f at P.
f(x) = x² - 5; P(3,4)

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 the limit \( f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h} \).
2. Step 2: Identify the function \( f(x) = x^2 - 5 \) and the point \( P(3, 4) \). We need to find the derivative \( f'(x) \) and evaluate it at \( x = 3 \).
3. Step 3: Substitute \( f(x) = x^2 - 5 \) into the derivative definition: \( f'(x) = \lim_{h \to 0} \frac{(x+h)^2 - 5 - (x^2 - 5)}{h} \).
4. Step 4: Simplify the expression inside the limit: \( (x+h)^2 - 5 - (x^2 - 5) = x^2 + 2xh + h^2 - 5 - x^2 + 5 = 2xh + h^2 \).
5. Step 5: Factor out \( h \) from the numerator: \( \frac{2xh + h^2}{h} = \frac{h(2x + h)}{h} = 2x + h \). Now, take the limit as \( h \to 0 \): \( f'(x) = \lim_{h \to 0} (2x + h) = 2x \). Evaluate \( f'(3) = 2(3) = 6 \).

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) = x² - 5, the derivative will provide the slope of the tangent line at any point on the graph.

Point of Tangency

The point of tangency is the specific point on the curve where the tangent line touches. In this case, point P(3,4) is where we need to evaluate the function and its derivative to find the slope of the tangent line. This point is essential for applying the derivative to determine the slope accurately.