Chapter 8 — Key Takeaways
A structured recap of implicit differentiation and related rates. Every idea in this chapter is the chain rule applied through a hidden dependency — either $y$ depending on $x$, or every quantity depending on time $t$. Hold onto that single sentence and the rest follows.
1. The Method of Implicit Differentiation (§8.2)
When a relation $F(x, y) = 0$ defines $y$ as a hidden function of $x$, find $\dfrac{dy}{dx}$ without solving for $y$:
- Differentiate both sides with respect to $x$.
- Apply the chain rule through $y$: every term containing $y$ produces a factor of $\dfrac{dy}{dx}$. A term $y^3$ differentiates to $3y^2\,y'$, not $3y^2$.
- Solve algebraically for $\dfrac{dy}{dx}$: collect the $y'$ terms, factor, divide.
The answer is a function of both $x$ and $y$ — and that is exactly the right amount of information, because a single point on the curve already knows which branch it is on.
Standard results worth recognizing
| Relation | $dy/dx$ | Note |
|---|---|---|
| $x^2 + y^2 = c$ | $-x/y$ | circle; tangent $\perp$ radius (§8.2) |
| $xy = c$ | $-y/x$ | rectangular hyperbola |
| $x^3 + y^3 = 6xy$ | $\dfrac{2y - x^2}{y^2 - 2x}$ | folium of Descartes (§8.2) |
| $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ | $-\dfrac{b^2 x}{a^2 y}$ | ellipse |
2. Tangents, Vertical Tangents, Second Derivatives
- Tangent line at $(x_0, y_0)$: $\;y - y_0 = y'(x_0, y_0)\,(x - x_0)$ (§8.4).
- Vertical tangents occur where the denominator of $y'$ vanishes while the numerator does not — features the explicit form $y = \pm\sqrt{\cdots}$ hides (§8.4).
- Higher derivatives (§8.3): differentiate $y'$ again, treating $y$ as a function of $x$ and substituting the known $y'$ back in. For the unit circle this yields $y'' = -1/y^3$; the sign reads off concavity directly.
3. Inverse-Function Derivatives (§8.5)
The cleanest route to any inverse derivative. From $y = f^{-1}(x) \Leftrightarrow x = f(y)$, differentiate implicitly:
$$\boxed{(f^{-1})'(x) = \frac{1}{f'\!\big(f^{-1}(x)\big)}}$$
This one identity generates the catalog:
$$\frac{d}{dx}\arctan x = \frac{1}{1+x^2}, \qquad \frac{d}{dx}\arcsin x = \frac{1}{\sqrt{1-x^2}}, \qquad \frac{d}{dx}\ln x = \frac{1}{x}.$$
Sign caution: when a square root appears, the inverse function's range dictates the sign. For $\arcsin$, the range $[-\tfrac{\pi}{2}, \tfrac{\pi}{2}]$ forces $\cos y \ge 0$, so the positive root is correct.
4. Logarithmic Differentiation (§8.6)
When $y$ is a tangle of products, quotients, and powers — or has a variable in both base and exponent — take $\ln$ first, then differentiate implicitly:
- $\ln y = \ln f(x)$.
- Differentiate: $\dfrac{y'}{y} = \dfrac{d}{dx}[\ln f(x)]$ (logs turn products into sums, powers into multipliers).
- $y' = y\cdot\dfrac{d}{dx}[\ln f(x)]$.
Signature example: $y = x^x \Rightarrow y' = x^x(\ln x + 1)$ — there is no honest way to get this without the log trick.
5. Related Rates: The Seven-Step Method (§8.7)
A related-rates problem is implicit differentiation with respect to time. Multiple quantities change together; a connecting equation binds them; differentiating it with respect to $t$ relates all their rates at once.
- Read and identify — what is changing, what rate is given, what rate is wanted.
- Draw and label — assign letters to changing quantities, not numbers.
- Record the given and wanted rates in symbols ($\dfrac{dx}{dt}$, $\dfrac{dy}{dt}$, …).
- Find the relating equation — Pythagoras, similar triangles, a volume formula, a trig ratio.
- Differentiate both sides with respect to $t$ — chain rule on every variable.
- Substitute the known values for the chosen instant and the known rates.
- Solve for the unknown rate, then check units and sign.
The golden rule: the geometry holds always; the numbers hold only now. Differentiate while everything is still a variable; freeze the numbers only at step 6.
Standard related-rate archetypes
| Setup | Relating equation | Worked in | Field |
|---|---|---|---|
| Sliding ladder | $x^2 + y^2 = L^2$ | §8.8 | safety/engineering |
| Conical tank filling | $V = \tfrac13\pi r^2 h$, $r = kh$ | §8.9 | chemical/civil eng. |
| Approaching/separating cars | $D^2 = x^2 + y^2$ | §8.10 | navigation, tracking |
| Walking shadow | similar triangles, $s = \tfrac12 x$ | §8.11 | optics/lighting |
| Filling trough | $V = (\text{cross-section})\times L$ | §8.12 | construction |
| Changing angle | $\tan\theta = h/d$ | §8.13 | radar/camera tracking |
| Inflating sphere | $V = \tfrac43\pi r^3$, $S = 4\pi r^2$ | §8.14 | balloons, imaging, biology |
Key identity from the sphere: $\dfrac{dV}{dt} = 4\pi r^2\dfrac{dr}{dt}$ — volume rate = surface area × radial rate (the shell, a glimpse of Chapter 18's integration).
6. Implicit Differentiation Beyond Geometry (§8.15)
A held-constant constraint binds variables implicitly. For a Cobb–Douglas isoquant $A\,L^\alpha K^\beta = Q_0$, implicit differentiation gives the marginal rate of technical substitution:
$$\frac{dK}{dL} = -\frac{\alpha K}{\beta L}.$$
Same gesture as the circle's $-x/y$ — a constraint differentiated implicitly. Physics relates state variables $(P, V, T)$ via $PV = nRT$ the same way.
7. The Two Most Common Errors
- Forgetting the chain-rule factor. Differentiating $y^3$ as $3y^2$ instead of $3y^2\,y'$. Tell-tale symptom: your answer for $y'$ contains no $y$, or $y'$ never appears in the differentiated equation. Every $y$-term must produce a $y'$.
- Substituting numbers before differentiating (related rates). Plugging in $x = 6$ early turns $x$ into a constant with $\dfrac{dx}{dt} = 0$, throwing away the motion. Differentiate first; substitute at step 6.
Secondary traps: dropping a held-constant quantity entirely (its derivative is zero, but its value still appears in the equation); sign/quadrant errors in inverse-trig square roots; forgetting to eliminate a second changing length (e.g. $r$ in the cone) via similar triangles before differentiating.
8. Skills Checklist
- [ ] Differentiate an implicit relation for $\dfrac{dy}{dx}$.
- [ ] Find tangent and vertical-tangent lines to implicit curves.
- [ ] Compute $y''$ implicitly and read off concavity.
- [ ] Derive inverse-trig and $\ln$ derivatives from implicit differentiation.
- [ ] Apply logarithmic differentiation to products/quotients/variable exponents.
- [ ] Execute the seven-step related-rates method on ladders, tanks, vehicles, shadows, and tracking angles.
- [ ] Check units and signs and interpret a negative rate physically.
9. Connections
- Chapter 7 — the chain, product, and quotient rules used in every computation here.
- Chapter 9 — curve analysis uses implicit $y'$ and $y''$ to locate tangents and concavity on non-function curves.
- Chapter 10 — optimization often differentiates a constraint implicitly to eliminate a variable.
- Chapter 19 — differential equations frequently produce solutions defined only implicitly, $F(x,y) = C$.
- Chapter 31 — optimization in several variables: the isoquant/Lagrange-multiplier formalism is the multivariable successor to §8.15's MRTS.
- Chapter 30 — the Implicit Function Theorem (§8.4 sidebar) and the gradient generalize implicit differentiation to many variables.
What's Next
Chapter 9 turns derivatives loose on the shape of functions — where they rise and fall, where they bend, where they peak — and several of those analyses lean on the implicit $y'$ and $y''$ you computed here. Chapter 10 then uses derivatives to find best-possible values, repeatedly calling on the implicit differentiation of constraints you practiced in §8.15.
Reflection
We did not invent a new kind of derivative. Implicit differentiation and related rates are the Chapter 7 chain rule, applied with the willingness to differentiate an equation we cannot solve and to let the answer involve more than one variable. A circle's tangent falls out as $-x/y$, a ladder's fall as $-\tfrac{x}{y}\dfrac{dx}{dt}$, an economy's substitution rate as $-\dfrac{\alpha K}{\beta L}$ — all the same gesture, aimed at different worlds. Calculus is the mathematics of change, and change almost never arrives pre-solved for $y$. Now you can differentiate it anyway.