Author Archives: hangtime
Finding Second Derivatives With Implicit Differentiation
The derivative of the derivative is the "second derivative", also known as d2y/dx2 or y'' (y double prime). Finding the second derivative is as easy as just taking the derivative of the first derivative, but in these implicit problems you end up having to keep plugging in stuff at each step if you want full credit. You'll see what I mean.
Finding Tangent Lines Using Implicit Differentiation
We covered tangent lines pretty well in the finding tangent lines chapter, but in this video we'll be using implicit differentiation to find the derivative (a.k.a. slope) of the tangent line. Then it's just plug-and-chug using point-slope form!
What The Heck Is Implicit Differentiation?
Glad you asked so politely. This is where you have an equation like xy-y=3 and they want you to find dy/dx, a.k.a. y' (y prime). Previously, we've only taken derivatives of functions with just one y or f(x); adding that second y makes these a whole new can of worms! These aren't too bad, you just have to be careful to NEVER forget y'.
In this chapter we'll cover the basics of taking derivatives implicitly (finding y'), using them to find equations of tangent lines, and finding second derivatives (y'').
Quotient Rule With Chain Rule
If you thought the product rule was bad, this is another level! Same organizational tricks will get you through it in one piece, though, so tune in for that.
Product Rule With Chain Rule
Title pretty much says it all. We'll get a look at differentiating (taking derivatives of) exponentials, trig functions, polynomials to powers, and natural logs, all multiplied together. The key to getting through this stuff is to be super-organized in your work, finding u' and v' separately before trying to plug them into the formula. Don't do this stuff in your head!
This chapter brings the chain rule to product and quotient derivatives. If you haven't, check out the first product & quotient rule chapter first.
Derivatives of Natural Logs (with Chain Rule)
Most Calculus teachers don't emphasize derivatives of natural logs. However, these gems are very common later in the semester when you're doing integrals, so I figure we might as well take a look at them now. Plus, they're a great way to practice the chain rule!
Derivatives of Exponentials (with Chain Rule)
Without the chain rule, it's pretty boring differentiating ex all the time, because it's always the same. But thanks the chain rule, your teacher will now demand that you take the derivatives of e with all kinds of crazy exponents, and it will be up to you to deliver.
Derivatives of Trig Functions (with Chain Rule)
We already saw how to differentiate trig functions as long as they only had an X or theta. Now we'll see how differentiate harder stuff like sin(3x), tan(x2+4), or even sec(ex).
The Power Rule (with Chain Rule)
In a previous derivatives chapter we already saw how to use the power rule to take derivatives of powers of x. Now we'll use the chain rule to take derivatives of anything -- polynomials, sine, cosine -- to a power, with the help of u'.
Hard Chain Rule Problems
These are just a bunch of really hard chain rule problems, where we'll have to use the chain rule two or three times on the same problem. These will help get you used to the most confusing aspect of the chain rule, which is figuring out when you're done once you're in two or three chain rules deep.
Derivatives of Roots & Radicals (with Chain Rule)
In a previous derivatives chapter we saw how to take derivatives of square roots and cube roots of x. Now, thanks to the chain rule, we'll take derivatives of all kinds of crazy roots and radicals.
What Is The Chain Rule? (free)
I'm not going to throw you any major calculus theory about where the chain rule comes from. That will just confuse you, and besides, I don't really know where this thing comes from. Instead, in this video I'll show you how to spot whether a problem needs the chain rule or not, because that's the crucial first step in any derivative problem. In later videos we'll get into how to actually use the chain rule in each problem type.
In this lengthy chapter we'll re-learn all the derivative formulas, except this time using the Chain Rule too: exponents (power rule), roots & radicals, trig functions, inverse trig functions, exponentials, and natural logs. A must-watch for Calculus students!
Quotient Rule (without chain rule)
I shouldn't even be showing you this formula, because my advice is to never use it if you can possibly avoid it. It's a mess. So in this video I'll not only show you how to use it, I'll show you how to algebraically manipulate things so that you can turn quotient problems into a product rule problem instead. All without the chain rule, obviously, 'cause that would make this monster even nastier.
Product Rule (without chain rule)
If you're taking Calculus, you've probably already seen the product rule, and you know it's a bit of a monster. But it gets a lot worse when you have to do the product rule AND the chain rule. So, in this video we'll practice easier product rule problems before things get crazy.