Taking derivatives : Khan Academy on a Stick

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

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.

Derivative as slope of a tangent line
Understanding that the derivative is just the slope of a curve at a point (or the slope of the tangent line)
Tangent slope as limiting value of secant slope example 1
Tangent slope as limiting value of secant slope example 2
Tangent slope as limiting value of secant slope example 3
Calculating slope of tangent line using derivative definition
Calculus-Derivative: Finding the slope (or derivative) of a curve at a particular point.
The derivative of f(x)=x^2 for any x
Calculus-Derivative: Finding the derivative of y=x^2
Formal and alternate form of the derivative
Formal and alternate form of the derivative for ln x
Formal and alternate form of the derivative example 1
Calculus: Derivatives 1
Finding the slope of a tangent line to a curve (the derivative). Introduction to Calculus.
Calculus: Derivatives 2
More intuition of what a derivative is. Using the derivative to find the slope at any point along f(x)=x^2

Introduction to derivatives

Discover what magic we can derive when we take a derivative, which is the slope of the tangent line at any point on a curve.

Derivative intuition module
Sal walking through the derivative intuition module made by Ben Eater
Interpreting slope of a curve exercise
Graphs of functions and their derivatives example 1
Where a function is not differentiable
Identifying a function's derivative example
Figuring out which function is the derivative
Intuitively drawing the derivative of a function
Intuitively drawing the antiderivative of a function
Visualizing derivatives exercise
Exercise available at https://www.khanacademy.org/math/calculus/differential-calculus/derivative_intro/e/visualizing_derivatives Exercise made by Stephanie Chang

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).

Using derivative properties

Derivative properties and intuition

Let's now get a better understanding of the different derivative-related notations and use them to better understand properties of derivatives.

Power rule
Is the power rule reasonable
Derivative properties and polynomial derivatives
Proof: d/dx(x^n)
Proof that d/dx(x^n) = n*x^(n-1)
Proof: d/dx(sqrt(x))
Proof that d/dx (x^.5) = .5x^(-.5)
Power rule introduction (old)
Determining the derivatives of simple polynomials.

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!

Chain rule

You can take the derivatives of f(x) and g(x), but what about f(g(x)) or g(f(x))? The chain rule gives us this ability. Because most complex and hairy functions can be thought of the composition of several simpler ones (ones that you can find derivatives of), you'll be able to take the derivative of almost any function after this tutorial. Just imagine.

Derivatives of sin x, cos x, tan x, e^x and ln x
The product rule for derivatives
Product rule for more than two functions
Quotient rule from product rule
Quotient rule for derivative of tan x
Using the product rule and the chain rule
Product rule
The product rule. Examples using the Product and Chain rules.
Quotient rule and common derivatives
Why the quotient rule is the same thing as the product rule. Introduction to the derivative of e^x, ln x, sin x, cos x, and tan x
Equation of a tangent line
Finding the equation of the line tangent to f(x)=xe^x when x=1

Product and quotient rules

You can figure out the derivative of f(x). You're also good for g(x). But what about f(x) times g(x)? This is what the product rule is all about. This tutorial is all about the product rule. It also covers the quotient rule (which really is just a special case of the product rule).

Implicit differentiation

Like people, mathematical relations are not always explicit about their intentions. In this tutorial, we'll be able to take the derivative of one variable with respect to another even when they are implicitly defined (like "x^2 + y^2 = 1").

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.

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".

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!