The Derivative and the Slope of a Graph

Tangent Lines and the Rate of Change

• The slope of a non-linear graph changes from point to point; the rate at which the graph rises or falls can be determined by the slope of the tangent line at a specific point.

• A tangent line to a graph at point $P(x_1, y_1)$ is the line that best approximates the slope of the graph at that point.

• Unlike a circle, where a tangent line intersects at only one point, a tangent line to a general function may intersect the graph at multiple points.

Slope and the Limit Process

• A secant line is used to approximate the slope of a tangent line by passing through the point of tangency and a second point on the graph.

• The slope of the secant line is expressed by the difference quotient: $m_{sec} = \frac{f(x + \Delta x) - f(x)}{\Delta x}$

• As $\Delta x \rightarrow 0$, the secant lines approach the tangent line, and the difference quotient provides the exact slope of the tangent line at $(x, f(x))$.

Definition and Notation of the Derivative

• The derivative of a function $f$ at $x$ is the limit of the difference quotient: $f'(x) = \lim_{\Delta x \rightarrow 0} \frac{f(x + \Delta x) - f(x)}{\Delta x}$

• Common notations for the derivative of $y = f(x)$ include:

  • $f'(x)$

  • $y'$

  • $\frac{dy}{dx}$

  • $\frac{d}{dx}[f(x)]$

• For example, if $f(x) = x^2 + 1$, the derivative formula is $f'(x) = 2x$.

• For the function $y = \frac{2}{t}$, the derivative is $\frac{dy}{dt} = -\frac{2}{t^2}$.

Differentiability and Continuity

• Differentiability Implies Continuity: If a function $f$ is differentiable at $x = c$, then $f$ is continuous at $x = c$.

• Continuity is a necessary but not sufficient condition for differentiability; a function can be continuous but not differentiable at a point.

• Factors that prevent differentiability at a point include:

  • Sharp turns in the graph.

  • Vertical tangent lines.

  • Points of discontinuity.