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.
Newton, Leibniz, and Usain Bolt
Why we study differential calculus.

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).
Interpreting slope of a curve exercise
Recognizing slope of curves
Identify intervals with positive and negative slope on graphs of functions
Graphs of functions and their derivatives example 1
In this problem, we are about the value of a function and its derivative based on its graph.
Where a function is not differentiable
Find where a function is not differentiable.
Identifying a function's derivative example
Given the graph of a function, we are asked to recognize the graph of its derivative.
Figuring out which function is the derivative
More practice with the relationship between the graph of a function and the graph of its derivative.
Graphs of functions and their derivatives
After watching Sal go through some examples identifying what the graph of a derivative might look like, I bet you want to practice some yourself, don't you?
Intuitively drawing the derivative of a function
Rather than merely identifying the graph of the derivative of a function, a true test of understanding is to draw that derivative from a blank slate.
Intuitively drawing the antiderivative of a function
No we go the other way from what we did in the last video, drawing the graph of a function given what its derivative looks like.
Visualizing derivatives exercise
Sal walks through an exercise made by Stephanie Chang, where you fit a portion of the antiderivative of a function with the graph of the function itself.
Visualizing derivatives
Practice visualizing the derivative.

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.
Chain rule introduction
An introductory example of the chain rule for the function (sin(x))^2
Chain rule definition and example
See the general formula for the chain rule, as well as an example of what applying this formula looks like in practice.
Chain rule for derivative of 2^x
Even though the function 2^x doesn't look like a composition at first, the chain rule can help us find its derivative.
Derivative of log with arbitrary base
Using the chain rule, we can find the derivative of any logarithm based on the derivative of ln(x)
Chain rule example using visual function definitions
Chain rule example using visual information
Chain rule on two functions
Practice applying the chain rule.
Chain rule with triple composition
Why compose just two functions?  Let's go crazy and compose three!
Derivative of triple composition
An example of the chain rule applied to a triple composition.
Chain rule on three functions
Solidify your understanding of the chain rule by practicing it on triple compositions.
Chain rule overview
A quick overview sheet of the chain rule.
A difficult but interesting derivative word problem

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.
Differentiability implies continuity
If a function has a derivative at all its points, it must be continuous.  See why!
Change in continuous function approaches 0
Change in continuous function approaches 0
Chain rule proof
Here we use the formal properties of continuity and differentiability to see why the chain rule is true.

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!
Proof: d/dx(ln x) = 1/x
Taking the derivative of ln x
Proof: d/dx(e^x) = e^x
Proof that the derivative of e^x is e^x.
Proofs of derivatives of ln(x) and e^x
Doing both proofs in the same video to clarify any misconceptions that the original proof was "circular".