Differentiation rules

The first principle of calculus is tedious and it gives you plenty of opportunities to make mistakes. Fortunately, there are some shortcuts.

Constant rule

This is really just a special case of the product rule, but you will use it so often that it’s easier to think of it separately. If k is a constant, then


Sum & difference rules

The derivative of the sum is equal to the sum of the derivatives. The same goes for subtraction. This means that


or, using Leibniz’s notation,


Power rule

Polynomials are very easy to differentiate thanks to the power rule.


For example, the derivative of x100 with respect to x is 100x99.

Product rule

The derivative of the product is not the product of the derivatives. Rather,


or, using Leibniz’s notation,


Quotient rule

We can use the product and power rules together to tackle rationals, but using the quotient rule is much easier:


or, using Leibniz’s notation,


Chain rule

To differentiate composite functions, we need to use the chain rule:


or, using Leibniz’s notation,