Chain, Product & Quotient Rules — Maths for Thermodynamics & Kinetics

Differentiating the awkward ones

Three rules for combinations of functions

The basic derivative rules handle simple functions in isolation. Once functions are combined — nested inside each other, multiplied together, or divided — you need three additional rules:

The chain rule handles a function of a function: $y = g (h (x))$ . Differentiate the outer function at the inner value, then multiply by the derivative of the inner. The mental picture is “peel the onion, layer by layer.”

The product rule handles two functions multiplied: $y = u (x) v (x)$ . The derivative is “derivative of the first times the second, plus the first times derivative of the second.” Neither factor alone captures how the product changes; both contribute.

The quotient rule handles a ratio $y = u / v$ . It is the product rule applied to $u \cdot v^{- 1}$ , rearranged into the form most people memorise: $(u^{'} v - u v^{'}) / v^{2}$ .

Chain rule

\frac{d}{d x} g (h (x)) = g^{'} (h (x)) \cdot h^{'} (x)

Product rule

(u v)^{'} = u^{'} v + u v^{'}

Quotient rule

(\frac{u}{v})^{'} = \frac{u ^{'} v - u v ^{'}}{v ^{2}}

Arrhenius chain rule

\frac{d k}{d T} = \frac{E _{a}}{R T ^{2}} k

outer e^u, inner −Ea/RT

In physical chemistry

Differentiating

e^{- k t}

(rate laws),

A e^{- E_{a} / RT}

(Arrhenius), and Gaussian peaks all lean on the chain rule. Rate expressions that are ratios — like the Michaelis–Menten rate

v = V_{m a x} [S] / (K_{M} + [S])

— need the quotient rule. Open the rule explorer and compare

f

with

f^{'}

A key application of the chain rule in physical chemistry is differentiating the Arrhenius equation with respect to reciprocal temperature $1/ T$ . Taking the natural log first gives $ln k = ln A - \frac{E _{a}}{R} \cdot \frac{1}{T}$ , a straight line when plotted as $ln k$ versus $1/ T$ . Differentiating directly with respect to $1/ T$ yields slope $- E_{a} / R$ — the foundation of the van't Hoff plot and the experimental determination of activation energies. Equivalently, by the chain rule with $u = 1/ T$ : $\frac{d ln k}{d ( 1/ T )} = - \frac{E _{a}}{R}$ .

The quotient rule gives a physically important result when applied to the Michaelis–Menten velocity. At low substrate concentrations ( $[S] ≪ K_{M}$ ) the derivative $d v / d [S] \approx V_{m a x} / K_{M}$ — a constant, so the rate is approximately linear in $[S]$ (apparent first order). At saturation ( $[S] ≫ K_{M}$ ) the derivative approaches zero — the enzyme is fully occupied and additional substrate produces no extra rate.

d ln k / d(1/T)

\frac{d ln k}{d ( 1/ T )} = - \frac{E _{a}}{R}

— slope of the Arrhenius (van't Hoff) plot.

d ln k / dT

Chain rule:

\frac{d ln k}{d T} = \frac{E _{a}}{R T ^{2}}

. Always positive.

Product rule mnemonic

“d-first times second, plus first times d-second”:

(uv)^{'} = u^{'} v + u v^{'}

Quotient rule — sign order

Top:

u^{'} v - u v^{'}

(not

u v^{'} - u^{'} v

). Denominator:

v^{2}

Common pitfalls

Forgetting the inner derivative (chain rule). Writing $\frac{d}{d t} e^{- k t} = e^{- k t}$ misses the factor of $- k$ . The correct result is $- k e^{- k t}$ . Every layer inside the exponential contributes a factor.
Mixing up product rule and chain rule. For $f = x e^{- x}$ (two factors multiplied) use the product rule. For $g = e^{- x^{2}}$ (one function inside another) use the chain rule. They are different operations — check whether you have a product or a composition.
Quotient-rule sign/order error. The numerator is $u^{'} v - u v^{'}$ , not $u v^{'} - u^{'} v$ . Reversing the sign flips the answer. Write out $u, v, u^{'}, v^{'}$ explicitly before combining.
Differentiating a constant as if it depended on the variable. In $k = A e^{- E_{a} / RT}$ , when differentiating with respect to $T$ , the pre-exponential $A$ and $E_{a} / R$ are constants — do not differentiate them.

Rule explorer

Switch between chain, product and quotient rules — the solid curve is f and the dashed curve is f′.

Rule:

f(x) f′(x)

Rule

\frac{d}{d x} g (h (x)) = g^{'} (h) h^{'} (x)

Example

f (x) = e^{- x^{2}}

Derivative

f^{'} (x) = - 2 x e^{- x^{2}}

A Gaussian — the inner −x² makes this need the chain rule. Same structure as a spectroscopic peak. Notice f′ crosses zero exactly where f has a turning point.

Worked example 1Chain rule on the first-order decay exponential

A first-order reaction gives $[A] = 0.80 e^{- 0.050 t} M$ ( $t$ in s). Find $d [A] / d t$ and evaluate the rate of disappearance at $t = 20 s$ .

Worked example 2Quotient rule on the Michaelis–Menten rate

Enzyme kinetics gives $v = \frac{V _{m a x} [ S ]}{K _{M} + [ S ]}$ . Differentiate with respect to $[S]$ to find how the rate changes with substrate concentration.

Worked example 3Chain rule on the Arrhenius equation

Differentiate $k = A e^{- E_{a} / RT}$ with respect to temperature $T$ and show the result can be written $d k / d T = E_{a} k / (R T^{2})$ .

Worked examples

Composite functions from thermodynamics and kinetics — try each before revealing the solution.

Worked example 4Product rule on a pre-exponential term

A modified rate expression for unimolecular dissociation is $k (T) = C T^{1/2} e^{- E_{a} / RT}$ . Differentiate with respect to $T$ .

Worked example 5Chain rule on a Gaussian spectroscopic peak

A Gaussian absorption peak is modelled as $A (ν) = A_{0} exp (- \frac{( ν - ν _{0} ) ^{2}}{2 σ ^{2}})$ . Differentiate with respect to $ν$ and find where $d A / d ν = 0$ .

Worked example 6Product rule on heat-capacity enthalpy

The sensible-heat enthalpy increment is $H (T) = \int C_{p} (T) d T$ , but in differential form the quantity $C_{p} (T) \cdot T$ appears in some entropy expressions. Differentiate $f (T) = C_{p} (T) \cdot T$ with respect to $T$ , where $C_{p} (T) = a + b T$ (Shomate linear approximation, $a = 28.0$ , $b = 0.030 J mo l^{- 1} K^{- 2}$ ).

Worked example 7d(ln k)/d(1/T) from the Arrhenius equation

Show that taking the natural log of $k = A e^{- E_{a} / RT}$ and differentiating with respect to $1/ T$ gives slope $- E_{a} / R$ , explaining why a plot of $ln k$ vs $1/ T$ is linear.

ChallengeSecond derivative of the Arrhenius k — curvature of the van't Hoff plot

Starting from $k = A e^{- E_{a} / RT}$ , you derived $d k / d T = E_{a} k / (R T^{2})$ (chain rule).

(a) Differentiate again to find $d^{2} k / d T^{2}$ in terms of $k$ , $E_{a}$ , $R$ and $T$ .
(b) Show that the equivalent result in terms of $ln k$ gives $d ln k / d T = E_{a} / (R T^{2})$ and explain its sign.

Check yourself

Four quick questions on chain, product and quotient rules.

Question 1 of 4 · Score 0

The chain rule differentiates a 'function of a function'. For y = g(h(x)) it gives:

\frac{d y}{d x} = g^{'} (h (x)) \cdot h^{'} (x)

Choose an answer.