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From: Re: Effective Tax Rate [How to Calculate? ], intended for Marginal Tax Rate

Marginal rates explained
In finance, the term "marginal" means derivative. Understanding calculus can give insight into concepts like marginal demand and marginal supply.

Applying calculus principles, the equivalence of marginal and derivatives is a matter of definition as described below:

Your income increases from $20,000 to $20,100; which provides you with $100 of extra income. This additional $100 of income results in $15 of additional tax. The marginal rate is therefore 15% = $15 / $100. Stated another way, your marginal tax rate at an income of $20,000 is 15%.

$$ \begin{align} {marginal\ tax\ rate\ at\ income\ of\ $20,000} & = \frac{tax(20,000 + 100) - tax(20,000)}{(20,000 + 100) - (20,000)} \\ \\ {marginal\ tax\ rate(I)}&= \frac{tax(I + \Delta) - tax(I)}{\Delta}\\ \\ \frac{df(x)}{dx}&=\lim_{\delta\rightarrow \infty} \frac{(f(x+\delta)-f(x))}{\delta} \end{align} $$

A partial derivative indicates that the tax is not a function of one income, e.g. total income, but several different types of income. For example, wages and capital gains. So we have not a function of one variable tax(income) but a function of two variables tax(wages,gains).

A partial derivative of a function of two or more variables is just the derivative if we change one variable and leave the others constant. This is how marginal tax rates are usually calculated. Refer to the previous example. At an income of $20,000, the marginal tax rate on wages is 15% because an extra $100 of wages increase taxes by $15. However, the marginal tax rate on capital gains is 0% because an extra $100 of capital gains results in $0 extra tax.

Putting these concepts together, let's convert some of the wages into capital gains (through various investments). Both wages and capital gains are a function of a third variable, the proportion chosen to allocate to gains, x, which results in a form of tax(wage(x),gain(x)). Looking how this tax changes as the allocation changes results in a full derivative in the case of tax with respect to x (formally as  &part;(tax(x))/&part;x).