Differential calculus

How would you like to follow in the footsteps of Euclid and Archimedes? Would you like to be able to determine precisely how fast Usain Bolt is accelerating exactly 2 seconds after the starting gun? Differential calculus deals with the study of the rates at which quantities change. It is one of the two principal areas of calculus. The fathers of modern calculus, Isaac Newton and Gottfried Leibniz, independently formulated the fundamental theorem of calculus relating differentiation and integration. Differential calculus and integral calculus are connected by the fundamental theorem of calculus, which states that differentiation is the reverse process to integration.
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Taking derivatives

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

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!

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.

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!