# Taking derivatives

Contents
Calculating derivatives. Power rule. Product and quotient rules. Chain Rule. Implicit differentiation. Derivatives of common functions.

## Introduction to differential calculus

The topic that is now known as "calculus" was really called "the calculus of differentials" when first devised by Newton (and Leibniz) roughly four hundred years ago. To Newton, differentials were infinitely small "changes" in numbers that previous mathematics didn't know what to do with. Think this has no relevence to you? Well how would you figure out how fast something is going *right* at this moment (you'd have to figure out the very, very small change in distance over an infinitely small change in time)? This tutorial gives a gentle introduction to the world of Newton and Leibniz.

## Using secant line slopes to approximate tangent slope

The idea of slope is fairly straightforward-- (change in vertical) over (change in horizontal). But how do we measure this if the (change in horizontal) is zero (which would be the case when finding the slope of the tangent line. In this tutorial, we'll approximate this by finding the slopes of secant lines.

## Visualizing graphs of functions and their derivatives

You understand that a derivative can be viewed as the slope of the tangent line at a point or the instantaneous rate of change of a function with respect to x. This tutorial will deepen your ability to visualize and conceptualize derivatives through videos and exercises. We think you'll find this tutorial incredibly fun and satisfying (seriously).

## Power rule

Calculus is about to seem strangely straight forward. You've spent some time using the definition of a derivative to find the slope at a point. In this tutorial, we'll derive and apply the derivative for any term in a polynomial. By the end of this tutorial, you'll have the power to take the derivative of any polynomial like it's second nature!

## Derivatives of common functions

For your derivative-taking-toolbelt, it's good to know the derivative of several common functions such as sin(x), tan(x), e^x, etc.

## Chain rule

If you know the derivatives of f(x) and g(x), the chain rule gives you the power to find the derivative of f(g(x)) or g(f(x)). Believe me, this is no small power. Most complex and hairy functions can be thought of the composition of several simpler ones (ones that you can find derivatives of), so after this tutorial, you'll be able to take the derivative of almost any function.

## Proving the chain rule

We've already been using the chain rule, but let's take a moment to really convince ourselves that it'll always work. It's not necessary to watch these videos before moving on, but if you have the time, they'll help to deepen your understanding of derivatives, continuity, and the chain rule.

## Product and quotient rules

If you know the derivative of two functions f(x) and g(x), can you find the derivative of f(x)g(x)? What about f(x)/g(x)? Well, you this is exactly what you'll be able to do after this tutorial.

## Implicit differentiation

Some curves, like the circle "x^2+y^2 = 1", are not defined as the graph of some function, as in "y = f(x)". After all, mathematical relations, like people, are not always explicit about their intentions. Here we learn how to find the slopes of these implicitly defined curves.

## Derivatives of inverse functions

In this tutorial we explore a common method to find the derivatives of inverse tangent (arctangent), inverse sine (arcsine), inverse cosine (arccosine) and the natural logarithm function.

## Proofs of derivatives of common functions

We told you about the derivatives of many functions, but you might want proof that what we told you is actually true. That's what this tutorial tries to do!