Notes on the Constant Multiple Rule

Introduction to the Constant Multiple Rule

  • The session addresses the constant multiple rule for derivatives, a topic previously introduced but not proven in lecture by Professor Jerison.

  • Aim: Provide a proof and geometric intuition for the constant multiple rule.

Constant Multiple Rule Definition

  • Definition: For a differentiable function $f(x)$ and a constant $c$, the derivative of the function $c imes f(x)$ is given by:

    • ddx(c⋅f(x))=c⋅ddx(f(x))\frac{d}{dx}(c \cdot f(x)) = c \cdot \frac{d}{dx}(f(x))

Example

  • Consider the constant $c = 3$ and the function $f(x) = x^2$:

    • Applying the rule results in:

    • ddx(3x2)=3⋅ddx(x2)\frac{d}{dx}(3x^2) = 3 \cdot \frac{d}{dx}(x^2)

  • Therefore, the derivative becomes:

    • ddx(3x2)=3⋅2x=6x\frac{d}{dx}(3x^2) = 3 \cdot 2x = 6x

  • Implication: This shows that we do not need a special rule to compute derivatives of multiples of powers of $x^2$; we can use the rule of derivatives of powers instead.

Benefits of the Rule

  • Reduces the complexity of computations when dealing with derivative calculations involving constants.

  • Minimizes the necessity to revert to the limit definition of derivatives.

Proof of the Constant Multiple Rule

  • Step 1: Definition of the derivative:

    • The derivative of $c imes f(x)$ can be expressed as:

    • ddx(c⋅f(x))=lim⁡Δx→0c⋅f(x+Δx)−c⋅f(x)Δx\frac{d}{dx}(c \cdot f(x)) = \lim_{\Delta x \to 0} \frac{c \cdot f(x + \Delta x) - c \cdot f(x)}{\Delta x}

  • Step 2: Factor out the constant $c$:

    • =c⋅lim⁡Δx→0f(x+Δx)−f(x)Δx= c \cdot \lim_{\Delta x \to 0} \frac{f(x + \Delta x) - f(x)}{\Delta x}

  • Step 3: Recognizing the limit as the derivative definition:

    • The limit expression is the definition of the derivative of $f(x)$:

    • Thus, we derive:

    • ddx(c⋅f(x))=c⋅ddx(f(x))\frac{d}{dx}(c \cdot f(x)) = c \cdot \frac{d}{dx}(f(x))

  • This completes the proof of the constant multiple rule.

Geometric Interpretation

  • Example with $c=2$:

    • Consider the graphs of $y = f(x)$ and $y = 2f(x)$.

    • The graph of $y = 2f(x)$ is a vertical stretch of $y = f(x)$ by a factor of 2.

Visualization of Changes in the Graph

  • If the graph of $y = f(x)$ passes through the origin, so does $y = 2f(x)$.

  • For points above the X-axis in $f(x)$, the corresponding points in $2f(x)$ are twice their height; for points below, they are twice as low.

Understanding Derivatives Geometrically

  • The derivative represents:

    • The slope of the tangent line at any point.

    • It can be understood as the limit of the slopes of secant lines.

  • Comparing slopes of tangent lines:

    • For two curves $y=f(x)$ and $y=2f(x)$:

    • Choose points $x$ and x+Δxx + \Delta x on both curves.

    • The slopes of the secant lines through these points will be:

    • Same run (change in x, or Δx\Delta x).

    • The rise for $y = 2f(x)$ is double that of $y = f(x)$.

    • Therefore, the slopes of the tangent lines will also maintain this doubling:

    • Derivative of $y = 2f(x)$ is precisely double that of $y = f(x)$.

  • This agrees with the constant multiple rule both algebraically and graphically, confirming its validity.