Chain rule and product rule

Rules that build derivatives

The derivative measures how fast a function changes: $f'(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}$. You rarely compute that limit by hand. Instead you combine a few known derivatives ($(x^n)'=nx^{n-1}$, $(e^x)'=e^x$, $(\ln x)'=\frac{1}{x}$, $(\sin x)'=\cos x$) with rules for sums, products, quotients and compositions.

The product rule applies when two changing quantities multiply. The value of a position is price times quantity, $V=P\cdot N$; if both move, $\frac{dV}{dt}=P'N+PN'$: one part from the price changing, one from the quantity changing.

The chain rule applies when one function sits inside another: change flows from the inside out. A discount factor $D(r)=e^{-rT}$ depends on the rate $r$ through the exponent, so $\frac{dD}{dr}=e^{-rT}\cdot(-T)=-T\,e^{-rT}$: each unit increase in the rate cuts the factor by $T$ times its value.

• Sum rule: $(u+v)'=u'+v'$.
• Product rule: $(uv)'=u'v+uv'$.
• Quotient rule: $\left(\frac{u}{v}\right)'=\frac{u'v-uv'}{v^2}$.
• Chain rule: differentiate the outside, then multiply by the derivative of the inside.

Lab · what's graded

• Product rule20 pts
• Chain rule with a logarithm20 pts
• Chain rule with a sine15 pts
• Rules and formulas15 pts
• A derivative at a point15 pts
• A chain-rule derivation15 pts