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.