Differentiation Rules, Implicit Differentiation, and Related Rates
The algebra of derivatives - product, quotient and chain rules - plus differentiating relations implicitly and linking rates that move together.
Leads to: The chain rule reappears as Itô’s lemma (Phase 9); implicit differentiation underlies option greeks and yield curves.
Learning Objectives
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- Apply the power, product, quotient, and chain rules to differentiate compositions.
- Differentiate implicitly a relation that does not isolate y and solve for dy/dx.
- Set up and solve a related-rates problem by differentiating a constraint with respect to time.
- Verify a symbolic derivative numerically with a central difference.
Key Vocabulary
- Power rule
- d/dx x^n = n x^{n-1} for any real n.
- Product rule
- (fg)′ = f′g + fg′.
- Quotient rule
- (f/g)′ = (f′g − fg′)/g².
- Chain rule
- (f∘g)′(x) = f′(g(x))·g′(x); differentiate outer, keep inner, times inner′.
- Implicit differentiation
- Differentiating both sides of F(x,y)=0 treating y as a function of x, then solving for y′.
- Related rates
- Using a constraint to relate the time-derivatives of interdependent quantities.
The four rules you compose everything from
With four rules you can differentiate essentially any elementary formula. The subtle one is the chain rule: a composition’s rate is the outer rate times the inner rate.
Implicit differentiation
When a curve is given by a relation you cannot (or would rather not) solve for \(y\), differentiate both sides treating \(y=y(x)\) and using the chain rule on every \(y\) term, then solve for \(y'\).
Related rates
If two quantities satisfy a relation and both change with time, differentiating the relation with respect to \(t\) links their rates. Example: a spherical balloon with \(V=\tfrac43\pi r^3\) gives \(\frac{dV}{dt}=4\pi r^2\frac{dr}{dt}\), so a known inflation rate \(dV/dt\) determines how fast the radius grows.
Interactive: check a derivative numerically
- Forgetting the inner derivative in the chain rule - writing (e^{3x})′=e^{3x} instead of 3e^{3x}.
- Dropping the y′ factor when differentiating a y term implicitly (that factor is the chain rule).
- Mixing up the quotient-rule sign: the numerator is f′g − fg′, in that order.
- In related rates, plugging in specific numbers BEFORE differentiating, which freezes the variables.
- Differentiate the general relation first; substitute the instantaneous values only at the end.
- For nested compositions, peel from the outside in, multiplying each layer’s derivative.
- A central difference with h≈1e−5 is a cheap, reliable check on any hand derivative.
- Implicit differentiation gives greeks: differentiating a pricing relation yields sensitivities like delta and gamma.
Knowledge Check
Practical Exercise
A company’s revenue is \(R(q)=q\,p(q)\) where price depends on quantity via \(p(q)=100-2q\). (a) Find marginal revenue \(R'(q)\). (b) At what \(q\) is marginal revenue zero, and what does that mean?
(a) \(R(q)=q(100-2q)=100q-2q^2\), so by the power rule \(R'(q)=100-4q\). (You can also use the product rule on \(q\cdot p(q)\): \(R'=p(q)+q p'(q)=(100-2q)+q(-2)=100-4q\).)
(b) Set \(R'(q)=0\): \(100-4q=0\Rightarrow q=25\). Marginal revenue zero means selling one more unit adds no revenue - the revenue curve peaks here (a maximum, since \(R''=-4\lt 0\)). Beyond \(q=25\) the price cut outweighs the extra volume.
Lesson Summary
Formula Sheet Additions
Retrieval Practice
Close the lesson and answer from memory before checking. This is deliberate, effortful recall - the single highest-yield study action.
A: (f∘g)′(x)=f′(g(x))g′(x): differentiate the outer function at the inner value, then multiply by the inner derivative. The common error is omitting the inner derivative g′(x).
A: Write a relation among the quantities, differentiate it with respect to time (chain rule on each variable), then substitute the given instantaneous values and solve for the unknown rate - substituting numbers only after differentiating.
Completion Checklist
- I can explain the core ideas in my own words
- I worked the derivations/examples by hand
- I completed the interactive workbench(es)
- I passed the knowledge check
Source References
This lesson synthesizes and paraphrases concepts from the sources below. No copyrighted text, problem sets, or solutions are reproduced. Return to the originals for full depth.
- Calculus, Vol. I (Tom Apostol, 2nd ed., 1967) foundational - Vol. I, Ch. 4-5 - Differentiation rules, the chain rule, and implicit differentiation.
- A Mind for Numbers (Barbara Oakley, 2014) foundational - chunking & practice - Turning the rules into automatic chunks through interleaved practice.