Derivative Rules and Rates of Change

Introduction to Derivatives

• The derivative of a function (f'(x)) represents the slope of the tangent line to the function's graph at a specific point.

Fundamental Derivative Rules

The Constant Rule

• If a function f(x) is a constant, meaning f(x) = c, its graph is a horizontal line.

• The slope of a horizontal line is always zero.

• Therefore, the derivative of any constant is zero.

  • Formula: If f(x) = c, then f'(x) = 0.

  • Examples:

    • If y = 7, then y' = 0.

    • If f(x) = 0, then f'(x) = 0.

    • If s(t) = -3, then s'(t) = 0.

    • If y = k
      pi^2 (where k is a constant), then y' = 0.

  • Trick Question Alert: If f(x) = \pi^3, the derivative f'(x) = 0, because \pi^3 is a numerical constant, not a variable raised to a power.

Linear Functions

• If a function is a linear equation f(x) = mx + b, its graph is a straight line.

• The derivative represents the slope, so for a linear function, the derivative is simply the slope of the line.

  • Formula: If f(x) = mx + b, then f'(x) = m.

  • Example: For f(x) = x (where m=1, b=0), f'(x) = 1.

The Power Rule

• The power rule applies when a variable is raised to a power.

• Formula: If f(x) = x^n (where n is a rational number), then f'(x) = nx^{n-1}.

• Proof (briefly mentioned): Uses the binomial theorem and the limit definition of the derivative.

• Conditions for Differentiability at x=0:

  • If n is a positive integer, the function is differentiable at x=0.

  • If n is a negative integer (e.g., x^{-1} = 1/x), then the function and its derivative are undefined at x=0.

  • If n is a fractional exponent (e.g., x^{1/3} = \sqrt[3]{x}), the derivative might be undefined at x=0 even if the original function is defined. This often indicates a vertical tangent line at that point, meaning the slope is infinite and the derivative does not exist.

• Examples:

  • If f(x) = x^3, then f'(x) = 3x^2.

  • If f(x) = x^{1/3} (cube root of x), then f'(x) = \frac{1}{3}x^{-2/3} = \frac{1}{3x^{2/3}} = \frac{1}{3\sqrt[3]{x^2}}. (Note: function is defined at x=0, but derivative is not; this implies a vertical tangent at x=0).

  • If y = \frac{1}{x^2} = x^{-2}, then y' = -2x^{-3} = -\frac{2}{x^3}. (Undefined at x=0).

Derivatives of Trigonometric Functions

• Derivative of sine: If f(x) = \sin x, then f'(x) = \cos x.

• Derivative of cosine: If f(x) = \cos x, then f'(x) = -\sin x.

  • (Proof is similar to other basic derivative proofs but takes a few minutes, not covered in detail here).

The Constant Multiple Rule

• If a function is multiplied by a constant, the derivative of the product is the constant multiplied by the derivative of the function.

• Formula: If g(x) = c \cdot f(x), then g'(x) = c \cdot f'(x).

  • Alternatively: If y = c x^n, then \frac{dy}{dx} = c(nx^{n-1}).

• Examples:

  • If y = 5x^3, then y' = 5(3x^2) = 15x^2.

  • If y = 2x^{-1}, then y' = 2(-1x^{-2}) = -2x^{-2} = -\frac{2}{x^2}.

  • If y = \frac{4t^2}{5} = \frac{4}{5}t^2, then y' = \frac{4}{5}(2t) = \frac{8}{5}t.

  • If y = 2\sqrt{x} = 2x^{1/2}, then y' = 2\left(\frac{1}{2}x^{-1/2}\right) = x^{-1/2} = \frac{1}{\sqrt{x}}.

The Sum and Difference Rules

• The derivative of a sum or difference of functions is the sum or difference of their individual derivatives.

• Sum Rule: If h(x) = f(x) + g(x), then h'(x) = f'(x) + g'(x).

• Difference Rule: If h(x) = f(x) - g(x), then h'(x) = f'(x) - g'(x).

• These rules are straightforward and apply naturally.

• Example: If h(x) = x^4 + x^3, then h'(x) = 4x^3 + 3x^2.

Product Rule (Crucial Distinction)

• The derivative of a product is NOT the product of the derivatives.

  • Demonstration: If h(x) = x^3 \cdot x^4 = x^7, then h'(x) = 7x^6.

  • However, if you incorrectly take f'(x) = (x^3)' = 3x^2 and g'(x) = (x^4)' = 4x^3, then f'(x) \cdot g'(x) = (3x^2)(4x^3) = 12x^5, which is not 7x^6.

• Derivation of the Product Rule (using adding zero):

  • To find the derivative of h(x) = f(x)g(x), we use the limit definition:
    h'(x) = \lim_{\Delta x \to 0} \frac{f(x + \Delta x)g(x + \Delta x) - f(x)g(x)}{\Delta x}

  • Add and subtract f(x+\Delta x)g(x) (a form of zero) in the numerator:
    h'(x) = \lim_{\Delta x \to 0} \frac{f(x + \Delta x)g(x + \Delta x) - f(x + \Delta x)g(x) + f(x + \Delta x)g(x) - f(x)g(x)}{\Delta x}

  • Factor f(x + \Delta x) from the first two terms and g(x) from the last two:
    h'(x) = \lim_{\Delta x \to 0} \left[ f(x + \Delta x) \frac{g(x + \Delta x) - g(x)}{\Delta x} + g(x) \frac{f(x + \Delta x) - f(x)}{\Delta x} \right]

  • As \Delta x \to 0:

    • f(x + \Delta x) \to f(x)

    • \frac{g(x + \Delta x) - g(x)}{\Delta x} \to g'(x)

    • \frac{f(x + \Delta x) - f(x)}{\Delta x} \to f'(x)

• Product Rule Formula: If h(x) = f(x)g(x), then h'(x) = f(x)g'(x) + g(x)f'(x).

  • Mnemonic: "First times derivative of the second, plus second times derivative of the first."

• Example application (implied, not fully worked out): To find the derivative of x \sin x, use f(x)=x and g(x)=\sin x.

Quotient Rule

• The quotient rule applies to functions that are ratios of two other functions.

• Formula: If h(x) = \frac{f(x)}{g(x)}, then h'(x) = \frac{g(x)f'(x) - f(x)g'(x)}{[g(x)]^2}.

  • Mnemonic (common for "low d high minus high d low over low squared"): "Low dee High minus High dee Low over Low Low (or Low Squared)."

Rates of Change and Motion

Average vs. Instantaneous Rate of Change

• Average Rate of Change: The slope of the secant line between two points on a curve. Calculated as \frac{\Delta y}{\Delta x} = \frac{f(x2) - f(x1)}{x2 - x1}.

  • Example: Average velocity of a car traveling 100 miles in 2 hours is 50 mph, regardless of speed fluctuations.

• Instantaneous Rate of Change: The slope of the tangent line at a single point on a curve. This is given by the derivative.

  • Calculus allows us to find instantaneous rates, unlike precalculus which focuses on average rates.

• Applications of rates of change: population growth, production rates, water flow rates, velocity, and acceleration.

Motion Along a Straight Line

• Used to describe the movement of a particle or object along a horizontal or vertical line.

• Position Function: s(t) gives the object's position at time t.

  • Delta s (\Delta s) is the change in position over a time interval delta t (\Delta t).

  • Average velocity over a short time interval: \text{Average Velocity} = \frac{\Delta s}{\Delta t} = \frac{s(t + \Delta t) - s(t)}{\Delta t}.

• Velocity Function: v(t) is the instantaneous rate of change of position, or the derivative of the position function.

  • Formula: v(t) = s'(t) = \lim_{\Delta t \to 0} \frac{s(t + \Delta t) - s(t)}{\Delta t}.

  • Velocity can be positive (moving right/up), negative (moving left/down), or zero (at rest).

• Speed: The absolute value of velocity.

  • Formula: \text{Speed} = |v(t)| = |s'(t)|.

  • Speed is always non-negative.

• Acceleration Function: a(t) is the rate of change of velocity, or the derivative of the velocity function (and the second derivative of the position function).

  • Formula: a(t) = v'(t) = s''(t).

Free-Fall Position Function (Standard Physics Formula)

• For an object moving purely under gravity (vertical motion), the position function is given by: s(t) = \frac{1}{2}gt^2 + v0t + s0

  • Where:

    • g is the acceleration due to gravity. (This will be negative in common physics applications as it pulls downwards).

    • Constants for g:

      • Metric system: g = -9.8 \text{ m/s}^2

      • English units: g = -32 \text{ ft/s}^2

    • v_0 (read as "v nought" or "v sub zero") is the initial velocity (velocity at time t=0).

    • s_0 (read as "s nought" or "s sub zero") is the initial position or initial height (position at time t=0).

• Example: Diver Jumping from a Platform

  • Initial height (s_0): 32 feet.

  • Initial upward velocity (v_0): 16 feet per second.

  • Position function (using g = -32 \text{ ft/s}^2):
    s(t) = \frac{1}{2}(-32)t^2 + 16t + 32
    s(t) = -16t^2 + 16t + 32

  • Question 1: When does the diver hit the water?

    • Set s(t) = 0 (height above water is zero).

    • -16t^2 + 16t + 32 = 0

    • Divide by -16: t^2 - t - 2 = 0

    • Factor: (t-2)(t+1) = 0

    • Solutions: t=2 or t=-1.

    • Since time cannot be negative in this context, the diver hits the water at t=2 seconds.

  • Question 2: What is the velocity at impact?

    • First, find the velocity function v(t) by taking the derivative of s(t):
      s'(t) = v(t) = \frac{d}{dt}(-16t^2 + 16t + 32)
      v(t) = -32t + 16 (constant term 32 becomes 0; 16t becomes 16; -16t^2 becomes -32t).

    • Now, plug in the impact time t=2 into v(t):
      v(2) = -32(2) + 16 = -64 + 16 = -48

    • The velocity at impact is -48 feet per second. The negative sign indicates the diver is moving downwards.