Solving Derivatives: All The Tricks and Techniques
In this page you’ll find everything you need to know about solving derivatives. My goal with this page is to make you a derivative solving machine.
I put the techniques you need to learn in an order that would make it easier for you to understand them. This page can be used as a map that can guide you in your study of derivatives, or you can use it to review all the techniques for solving derivatives.
1. The Essential Rules
1. Use The Definition
The most basic way of calculating derivatives is using the definition. This involves calculating a limit. To calculate derivatives this way is a skill.
As with any skill, you only improve with practice. We talk at length about how to use the definition on the page calculating the derivative by definition.
2. The Chain Rule
The chain rule is the most important rule for taking derivatives. With it you’ll be able to find the derivative of almost any function.
To learn about the chain rule go to this page: The Chain Rule.
3. The Product Rule
The product rule allows you to find derivatives of functions that are products of other functions. It is a very useful technique, and one of the few formulas you should memorize in calculus.
To learn about the product rule go to this page: The Product Rule.
4. The Quotient Rule
The quotient rule is just an special case of the product rule, so you don’t need to memorize another formula. I’ll show you a method for solving derivatives of quotients using the product rule.
To learn about this method go to this page: The Quotient Rule.
5. Implicit Differentiation
Implicit differentiation allows you to find derivatives of functions expressed in a funny way, that we call implicit. The key is in understanding the chain rule.
To learn about implicit differentiation go to this page: Implicit Differentiation.
2. The Essential Formulas
1. Derivative of Trigonometric Functions
To start building our knowledge of derivatives we need some formulas. Two basic ones are the derivatives of the trigonometric functions sin(x) and cos(x). We first need to find those two derivatives using the definition.
With these in your toolkit you can solve derivatives involving trigonometric functions using other tools like the chain rule or the product rule.
To learn about derivatives of trigonometric functions go to this page: Derivatives of Trigonometric Functions.
2. Derivative of Exponential Functions
We add another function to the list of those we know how to take the derivative of. This one is really useful and pretty. To learn about the derivative of exponential functions, go to this page.
3. Derivative of Logarithms
We add another formula to our list using, once again, the definition of the derivative. The result is pretty amazing.
To learn about the derivative of the natural log go to this page: Derivative of ln(x).
4. Derivative of Inverse Trig Functions
Here we learn about the derivative of arcsin(x), arccos(x) and others. These formulas are pretty hard to memorize, so it is good to know how to prove them to yourself. Go to this page: derivative of inverse trig functions.
5. Other Frequently Asked Formulas
There are some formulas for derivatives that I get asked very often. These are:
- The derivative of tan(x): This one is not as well-known as the derivatives of sin(x) and cos(x). To calculate it we use the quotient rule.
- Derivative of the Inverse: How do you find the derivative of the inverse of a function, in general?
Derivative of an Integral: You need to know about integrals before seeing this. This question always comes up, and on this page we do a pretty good job of clearing all doubts.