Calculus
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Taking derivatives
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.
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.
Visualizing 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!
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
- Special derivatives
- Chain rule introduction
- Chain rule definition and example
- Chain rule with triple composition
- Chain rule for derivative of 2^x
- Derivative of log with arbitrary base
- Chain rule 1
- Extreme Derivative Word Problem (advanced)
- The Chain Rule
- Chain Rule Examples
- Even More Chain Rule
- More examples using multiple rules
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).
- Derivatives of sin x, cos x, tan x, e^x and ln x
- Special derivatives
- Applying the product rule for derivatives
- Product rule for more than two functions
- Product rule
- Quotient rule from product rule
- Quotient rule for derivative of tan x
- Quotient rule
- Using the product rule and the chain rule
- Product Rule
- Quotient rule and common derivatives
- Equation of a tangent line
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").
- Implicit differentiation
- Showing explicit and implicit differentiation give same result
- Implicit derivative of (x-y)^2 = x + y + 1
- Implicit derivative of y = cos(5x - 3y)
- Implicit derivative of (x^2+y^2)^3 = 5x^2y^2
- Finding slope of tangent line with implicit differentiation
- Implicit derivative of e^(xy^2) = x - y
- Derivative of x^(x^x)
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!