Chain Rule & Related Differentiation Techniques — Lecture 3.3

Chain Rule – Conceptual Overview

  • Differentiation technique for a composition of two (or more) functions.
  • Intuition: “Differentiate the outside while keeping the inside unchanged, then multiply by the derivative of the inside.”
  • Typical symbolic composition: h(x)=g\bigl(f(x)\bigr)

Algebraic Statement (Version 1)

  • If h(x)=g\bigl(f(x)\bigr) and both f and g are differentiable, then
    \boxed{\displaystyle h'(x)=g'\bigl(f(x)\bigr)\,f'(x)}
    (read “derivative of outside evaluated at the inside times derivative of the inside”).

Alternative Notation – “u-Substitution” (Version 2)

  • Set u=f(x) and rewrite y=g(u).
  • The chain rule appears as a product of differentials:
    \frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}.
  • Same result, but the intermediate variable u often clarifies multi–step chains.

Step-By-Step Procedure (Power-Inside example)

  1. Identify outer and inner functions.
  2. Differentiate the outer; do not touch the inner expression.
  3. Multiply by the derivative of the inner.
  4. Simplify and/or substitute back if you used u.

Common Missteps & Tips

  • Forgetting the inner derivative (most frequent error).
  • Dropping exponents one too many times—power rule only affects the outer power.
  • For roots, rewrite as rational powers (e.g. \sqrt{\bullet}=\bullet^{1/2}) before differentiating.
  • Negative powers follow the same rule; the sign stays with the coefficient you pull down.

Worked Examples (Algebraic)

1. Pure Power

  • h(x)=(4x-1)^5
    • Inner: f(x)=4x-1,\quad f'(x)=4
    • Outer: g(u)=u^5,\; g'(u)=5u^4
    • \displaystyle h'(x)=5(4x-1)^4\cdot4=\boxed{20(4x-1)^4}

2. Radical of a Cubic Polynomial

  • h(x)=\sqrt{4-9x^3}=(4-9x^3)^{1/2}
    • f(x)=4-9x^3,\; f'(x)=-27x^2
    • g(u)=u^{1/2},\; g'(u)=\tfrac12u^{-1/2}
    • h'(x)=\tfrac12(4-9x^3)^{-1/2}\cdot(-27x^2)
      =\boxed{-\dfrac{27x^2}{2\sqrt{4-9x^3}}}

3. Negative Exponent (Reciprocal of a Quadratic)

  • h(x)=(x^2+3x+8)^{-2}
    • u=x^2+3x+8,\; du/dx=2x+3
    • dy/du=-2u^{-3}
    • \displaystyle h'(x)=-2(2x+3)(x^2+3x+8)^{-3}

4. “What are f and g?”

  • Given h(x)=\sqrt{3-5x}, choose
    • g(x)=3-5x\quad(\text{inner})
    • f(x)=\sqrt{x}\quad(\text{outer})
  • Recognition of inner/outer is flexible as long as f(g(x)) reproduces h(x).

5. Half-Power of a Cubic Expression

  • f(x)=5\,(2x^3+x)^{1/2}-4
    • g(x)=2x^3+x,\; g'(x)=6x^2+1
    • Derivative:
      \displaystyle f'(x)=\frac{5}{2}(2x^3+x)^{-1/2}(6x^2+1)
    • Evaluate at x=-2 by direct substitution if required.

Evaluating a Derivative at a Point (Numeric Example)

  • Goal: \left.\dfrac{d}{dx}\,[g(f(x))]\right|_{x=4}.
  • Chain rule: g'(f(4))\,f'(4). Read required y-values and slopes directly off the graphs of f and g.

Using Graphs to Compute a Composite Derivative

  • Procedure:
    1. From the f graph, read f(4) and the slope f'(4).
    2. Find the x-position on the g graph equal to f(4).
    3. Read the slope g'\bigl(f(4)\bigr).
    4. Multiply to obtain \dfrac{d}{dx}[g(f(x))]\big|_{x=4}.
  • Example in transcript returned g'(0)=-1 and f'(4)=? which yielded the composite derivative after multiplication.

Abstract Table Example

Suppose

  • f(4)=9,\;f'(4)=2
  • g(9)=12,\;g'(9)=-6

Then for h(x)=g(f(x))
h'(4)=g'\bigl(f(4)\bigr)\,f'(4)=g'(9)\cdot2=(-6)(2)=\boxed{-12}.

(Variations in the transcript included other data pairs such as g'(1)=5,\;f(2)=1,\;g(2)=-3—the calculation pattern is identical.)

Real-World Application – Sensitivity of Compound Interest

  • Monthly compounding for 10 years at annual rate r\%.
  • Balance function:
    A(r)=500\Bigl(1+\frac{r}{1200}\Bigr)^{120}
    (120 months in 10 years).
  • Differentiate w.r.t. interest rate r:
    \begin{aligned}
    A'(r)
    &=500\cdot120\Bigl(1+\frac{r}{1200}\Bigr)^{119}\cdot\frac{1}{1200}\[6pt]
    &=50\Bigl(1+\frac{r}{1200}\Bigr)^{119}.
    \end{aligned}
  • Interpretation: A'(r) is the dollar change in ending balance per 1-percentage-point change in the interest rate.

Evaluations

  1. At r=5:
    A'(5)=50\Bigl(1+\tfrac{5}{1200}\Bigr)^{119}\approx\boxed{\$82.01\text{/percentage-point}}.
  2. At r=7:
    A'(7)\approx\boxed{\$99.90\text{/percentage-point}}.
    The higher the interest rate, the more sensitive the balance becomes to further changes.

Combining Product, Quotient, and Chain Rules

1. Product × Chain

  • h(x)=3x\,(x^2+5)^4
    • Product rule: h'=L'R+LR' where L=3x,\;R=(x^2+5)^4
    • L'=3
    • R'=4(x^2+5)^3\cdot2x=8x\,(x^2+5)^3
    • \displaystyle h'(x)=3\,(x^2+5)^4+24x^2\,(x^2+5)^3.
    • Often factored as 3(x^2+5)^3\bigl[(x^2+5)+8x^2\bigr].

2. (Quotient-Style) Example with Both Rules

  • Suppose h(x)=\dfrac{(6x+7)^3}{x^2+3}. (The transcript listed the pieces as L=(6x+7)^3 and H=x^2+3.)
  • Quotient rule combined with chain:
    h'(x)=\frac{(6x+7)^3\cdot2x-(x^2+3)\cdot3(6x+7)^2\cdot6}{(x^2+3)^2}.
  • Careful bookkeeping of each inner derivative (18 underlined in the transcript) prevents sign errors.

3. Further Derivative Identities Used

  • For g(x)=(x^2+5)^4: g'(x)=8x\,(x^2+5)^3.
  • For f(x)=6x+7: f'(x)=6, hence \dfrac{d}{dx}(6x+7)^3=3(6x+7)^2\cdot6.

Big-Picture Connections & Significance

  • Chain rule is foundational; appears whenever units convert (e.g.
    concentration vs. time) or when one process feeds into another.
  • In physics: relates angular and linear velocities, links position–time curves through parametric equations.
  • In economics & finance: sensitivity analysis (marginal change) as in the compound-interest example.
  • Ethically: accurate rate-of-change calculations prevent mis-communication of risk (e.g.
    mis-stating how small temperature changes affect reaction rates).

Quick Reference Summary

  • \displaystyle (g\circ f)'=g'\circ f\;\cdot f'.
  • Differential form: dy=\dfrac{dy}{du}\,du,\quad du=\dfrac{du}{dx}\,dx, so dy=\dfrac{dy}{du}\dfrac{du}{dx}\,dx.
  • Don’t drop the inner derivative!
  • Combine seamlessly with product, quotient, implicit, and higher-order derivatives.