### All About Derivatives – How to Calculate Them Easily

Being that taking derivatives is a cornerstone element of calculus, it’s certainly worth it to be able to compute them as quickly and as painlessly as possible. There are two main ways or forms in which derivatives are denoted: Leibniz’s notations and Lagrange’s notations. The former is written as a fraction, commonly dy/dx, with higher derivatives being expressed via exponents of the “d” values (which are just operators symbolizing the derivative), as in d2y/d2x.

The latter, also known as prime notation, is written with apostrophe marks the number of which determines the order of the derivative, as in f’ and f”. The first one is read “f-prime”, and is the derivative of the function “f”, while the second one is the second derivative of the function “f”. Apostrophe marks are generally limited to three, after which integer superscripts enclosed in parenthesis are often used f(8).

There are many ways in which to find derivatives, but I’ve put together some of the main methods and equations which I believe can be used for most usual cases. Perhaps you’ve heard of some or all of them.

• Power Rule
• Product Rule
• Quotient Rule
• Derivatives of trigonometric functions

Power Rule

This rule covers any function with an exponent, as x2:

Product Rule

This rule covers any function that is a product of two functions, as (fg):

Quotient Rule

This rule covers any function that is a quotient of two functions:

Derivatives of Trigonometric Functions

There are two main trigonometric functions you should memorize, from which the rest can be based. These are the sine and cosine functions. You can remember these two by the phrase “sine keeps its sign when you differentiate“.

The derivative for the tangent function can be derived by simply dividing the sine over the cosine as per the trigonometric definition, then using the quotient rule to substitute for the final answer, as:

Here, we can see the quotient rule in action. The second step in the example above shows a change in signs due to the equations governing the derivatives of sines and cosines (see above). The fourth step shows that the sum of the squares of the sine and cosine of any function will add to 1. And if you’re familiar with trigonometry you’d know that secant is cosine’s reciprocal, 1 over cosine.

In short, the derivative of the tangent of x is equal to the secant squared of x. In the example above, we can also see the individual pros and cons of Lagrange’s notation vs. Leibniz’s notation, with the former being much more convenient to use when describing derivatives as a function in itself.

Higher Order Derivatives

Finally, there are times when you need to calculate higher order derivatives, as mentioned further above. This is the exact same thing as finding the first derivative, you just do the same thing again. In fact, you can keep on going as many times as you want. However, there are cases where the derivative will reach zero, after which any higher derivatives will also be zero. For example:

In the above example, the first and second derivatives are taken using the power rule outlined above, while the third derivative is taken using the product rule. The fourth derivative is taken using the fundamental principle that the derivative of any constant is zero. The graph of a constant function is horizontal, and as such, the derivative or slope, must be zero.

Higher Order Derivatives of Trigonometric Functions

You should also know that when it comes to trigonometric functions, their derivatives cycle in fours. In other words, the fifth derivative of a trigonometric function is the same as its first derivative. This means that trigonometric derivatives can be divided by 4 and the remainder will determine the order of derivative. For example: