DV

Derivatives: Rules, Examples, and Techniques

Derivative Basics and Linearity

  • Derivative of a constant is zero. If f(x) is a constant, d/dx[f(x)] = 0. This can be remembered conceptually or via the power rule: a constant is x^0, so d/dx[x^0] = 0.
  • Power rule (basic): \frac{d}{dx} x^{n} = n x^{n-1}.
    • Example: \frac{d}{dx} x^2 = 2x, and more generally, \frac{d}{dx} x^3 = 3x^2.
  • Constant multiples: for a constant c, \frac{d}{dx} [c \, f(x)] = c \, f'(x).
    • Example (from the transcript): if you have a term like 3 x^2, the derivative is 3 · (2x) = 6x; the constant 3 factors out and is applied after differentiating the rest.
  • Derivative of a sum/difference (linearity): \frac{d}{dx} [f(x) + g(x)] = f'(x) + g'(x).
    • Example: if f(x) = x^3 + x^3, then f'(x) = 3x^2 + 3x^2 = 6x^2.
  • Practical note: keeping track of terms with parentheses can help prevent mistakes when differentiating more complex expressions.

Increasing/Decreasing and Critical Points

  • The sign of the derivative determines increasing or decreasing:
    • Increasing where f'(x) > 0.
    • Decreasing where f'(x) < 0.
    • Horizontal tangent where f'(x) = 0.
  • Example setup discussed in the transcript:
    • If f'(x) factors as ( (x-2)(x-4) ), then f'(x) has zeros at x = 2 and x = 4.
    • Sign pattern (test points) for x < 2, 2 < x < 4, x > 4 is typically +, -, + (based on the factors).
    • Therefore, f is increasing on (−∞, 2) ∪ (4, ∞) and decreasing on (2, 4). f'(2) = f'(4) = 0 (horizontal tangents).
  • Quick visualization tip (handy in exams): plot or sketch the derivative to infer intervals of increase/decrease on the original function.

Product Rule

  • Formula: if f(x) = g(x) h(x), then
    f'(x) = g'(x) \cdot h(x) + g(x) \cdot h'(x).
  • Why it’s needed: ( (fg)' \neq f' g' ) in general (example: f(x) = x, g(x) = x gives (x·x)' = 2x, but f'(x)g'(x) = 1·1 = 1).
  • Quick check using a simple example:
    • Let f(x) = x · x. Take g(x) = x, h(x) = x.
    • g'(x) = 1, h'(x) = 1, so f'(x) = 1·x + x·1 = 2x, which matches the known derivative of x^2.
  • Worked example from the transcript (product rule with a root):
    • Let f(x) = (√x) · x, where g(x) = √x = x^{1/2}, h(x) = x.
    • g'(x) = (1/2) x^{-1/2}, h'(x) = 1.
    • f'(x) = g'(x) h(x) + g(x) h'(x) = (1/2) x^{-1/2} · x + x^{1/2} · 1 = (1/2) x^{1/2} + x^{1/2} = (\tfrac{3}{2}) √x.
  • Practical note from the lecture: product rule is often easier to apply directly to polynomials and simple products; you’ll see trig functions later where product rule is essential.

Quotient Rule

  • Formula: if f(x) = ( \frac{g(x)}{h(x)} ) with h(x) ≠ 0, then
    f'(x) = \frac{g'(x)\,h(x) - g(x)\,h'(x)}{[h(x)]^2}.
  • Important caveat: order matters (numerator vs denominator). It’s not simply g' h' divided by h^2.
  • Intuition: the quotient rule can be derived from the product rule and chain rule (but the transcript notes this without a full derivation here).
  • Practical advice from the transcript:
    • If the denominator is a single term (no addition/subtraction), you can sometimes avoid the quotient rule by rewriting with negative exponents and using the power rule directly.
    • If possible, simplify before differentiating to reduce the need for the quotient rule.
  • Example from the transcript (illustrating the complexity of direct quotient-rule application): a quotient with denominator x^2 and numerator involving a sum root{x} + x, illustrating why simplification or a product-rule approach can be preferable in some cases.
  • Another strategic note: if the denominator is a constant, treat the whole expression as a constant multiple and differentiate the numerator alone:
    • If f(x) = [p(x)]/7, then f'(x) = p'(x)/7.

Simplification Before Differentiation

  • Key strategy: whenever possible, simplify expressions before differentiating to minimize the use of the quotient rule and to reduce algebraic clutter.
  • Example from the transcript:
    • Consider f(x) = (√x + x) / x^2.
    • Rewrite as: \frac{√x}{x^2} + \frac{x}{x^2} = x^{1/2 - 2} + x^{1 - 2} = x^{-3/2} + x^{-1}.
    • Differentiate term-by-term using the power rule:
      \frac{d}{dx} [x^{-3/2} + x^{-1}] = \left(-\frac{3}{2}\right) x^{-5/2} - x^{-2}.
  • If the denominator has no x-dependence (a constant in the denominator), treat as a constant multiple and differentiate the numerator alone, then divide by that constant after differentiation.
  • Practical takeaway: simplifying first often avoids heavy quotient-rule work and can make later steps (like taking a second derivative) much cleaner.

Worked Examples and Practice Strategies

  • Product-rule examples (summary):
    • Example 1: f(x) = x · x → f'(x) = 2x.
    • Example 2: f(x) = (√x) · x → f'(x) = (3/2)√x (as shown above).
  • Quotient-rule practice: set up numerator and denominator clearly, label which is g and which is h, and apply
    f'(x) = \frac{g'(x)\,h(x) - g(x)\,h'(x)}{[h(x)]^2}.
  • A note on a common simplification: when the denominator is a polynomial or simple expression with addition/subtraction, you often cannot avoid the quotient rule; in such cases, you may want to simplify or decompose first if possible.
  • A cautionary example from the transcript: take a quotient like ( \frac{p(x)}{q} ) with constant q; you can bring out 1/q and differentiate p(x) directly.

Real-World Connections: Distance, Velocity, and Acceleration

  • Distinguish between distance, velocity, and acceleration:
    • If s(t) is a position (distance) function, then
    • velocity: v(t) = s'(t).
    • acceleration: a(t) = v'(t) = s''(t).
  • Example from the transcript: an acceleration problem where you differentiate twice and substitute t = 1 to obtain a numerical value (the instructor mentions getting 2 in that example).
  • Practical point: understanding the chain of derivatives helps connect calculus to motion and rates of change in the real world.

Notation, Style, and Exam Mindset

  • Variable naming: the transcript emphasizes consistency but notes you may see x, t, or u as the variable in problems on quizzes or exams.
  • Function labeling: to keep track when using product/quotient rules, it helps to name the two factors explicitly (e.g., let f(x) = g(x) · h(x) or f(x) = g(x)/h(x)) so you can clearly write g', h', etc.
  • Strategy advice from the lecture:
    • Prefer eliminating the quotient rule by simplifying first when possible.
    • If forced to use the quotient rule, write the rule clearly and compute diagonal products carefully; many students find the diagonal method easier.
    • In some problems, breaking a quotient into simpler pieces (via algebraic manipulation or exponent rules) can dramatically reduce effort.
  • A practical course tip mentioned: as you advance, chain rule will come into play more often; the Monday session will introduce it (not covered yet in this transcript).

Key Formulas to Memorize (LaTeX)

  • Power rule:
    \frac{d}{dx} x^{n} = n x^{n-1}.

  • Constant multiple rule:
    \frac{d}{dx} [c \, f(x)] = c \, f'(x).

  • Sum rule:
    \frac{d}{dx} [f(x) + g(x)] = f'(x) + g'(x).

  • Product rule:
    (fg)' = f' g + f g'.

  • Quotient rule:
    \left( \frac{g}{h} \right)' = \frac{g' h - g h'}{h^{2}}.

  • Derivative of the square root (and equivalent power form):
    \frac{d}{dx} \sqrt{x} = \frac{1}{2} x^{-1/2} = \frac{1}{2\sqrt{x}}.

  • Derivative of a constant: \frac{d}{dx} c = 0.

  • Increasing/Decreasing sign interpretation (conceptual): if f'(x) > 0, f is increasing; if f'(x) < 0, f is decreasing; if f'(x) = 0, horizontal tangent.

  • Tangent line (point-slope form):
    y - y1 = m (x - x1), ext{ with } m = f'(x_1).

  • Acceleration and velocity relation: if s(t) is distance, then
    v(t) = s'(t),

    a(t) = v'(t) = s''(t).

Quick Reference: Summary of When to Use What

  • Use power rule and linearity for polynomials and simple powers.
  • Use product rule for products of two functions (including polynomials with roots, exponentials, etc.).
  • Use quotient rule for ratios where the denominator involves x (or a function of x) and cannot be simplified away.
  • Always attempt simplification before differentiating when possible (avoid unnecessary quotient-rule work).
  • For approximate or conceptual questions, consider sign charts and intervals to determine increasing/decreasing behavior.
  • Remember that differentiation rules extend to other variables (t, u, etc.) with the same underlying principles; stay consistent with the variable in use.

End-of-Notes: Quick Check and Practice

  • If asked to find where a function is increasing or decreasing, compute f'(x), find critical points (where f'(x) = 0 or undefined), and test intervals.
  • If a problem involves distance s(t), velocity v(t) is s'(t) and acceleration a(t) is s''(t). Plug in the given value (e.g., t = 1) to obtain numerical acceleration.
  • When faced with a complicated quotient, first look for algebraic simplifications or decompositions that could avoid the quotient rule altogether.
  • Practice: set up a few quick problems mirroring the transcript, such as:
    • Evaluate the derivative of f(x) = 3x^2 + 5x, and determine where f is increasing.
    • Compute the derivative of f(x) = (√x + x)/x^2 using both quotient rule and simplification, and compare results.
    • Differentiate f(x) = (√x) · (x) and verify with the product rule.