yego.me
💡 Stop wasting time. Read Youtube instead of watch. Download Chrome Extension

Logarithmic functions differentiation | Advanced derivatives | AP Calculus AB | Khan Academy


2m read
·Nov 11, 2024

Let's say that Y is equal to log base 4 of x^2 + x. What is the derivative of y with respect to x going to be equal to? Now, you might recognize immediately that this is a composite function. We're taking the log base 4 not just of x, but we're taking that of another expression that involves x.

So we could say, this thing in blue, that's U of x. Let me do that in blue. So this thing in blue that is U of x, U of x is equal to x^2 + x. It's going to be useful later on to know what U prime of x is. So that's going to be just going to use the power rule here, so 2x + 1.

BR that brought that two out front and decremented the exponent. The derivative with respect to x of x is 1. We could say the log base 4 of this stuff, well, we could call that a function V. We could say V of, well, if we said V of x, this would be log base 4 of x.

And then we've shown in other videos that V prime of x is going to be very similar if this was log base e or natural log, except we're going to scale it. So it's going to be 1 over the natural log of 4 times x. If this was V of, if V of x was just natural log of x, our derivative would be 1/x.

But since it's log base 4, and this comes straight out of the change of base formulas that you might have seen, and we have a video where we show this, but we just scale it in the denominator with this natural log of 4. Or you could think of scaling the whole expression by 1 over the natural log of 4.

But we can now use this information because y, this y can be viewed as V of V of, remember, V is the log base 4 of something. But it's not V of x; we don't have just an x here. We have the whole expression that defines U of x. We have U of x right there.

And let me draw a little line here so that we don't get those two sides confused. And so we know from the chain rule the derivative of y with respect to x. This is going to be, this is going to be the derivative of V with respect to U. Or we could call that V prime, V prime of U of x, let me do the U of x in blue, V prime of U of x times U prime of x.

Well, what is V prime of U of x? We know what V prime of x is. If we want to do V prime of U of x, we would just replace wherever we see an x with a U of x. So this is going to be equal to V prime of U of x, and you just view it as you're taking the derivative of the green function with respect to the blue function.

So it's going to be 1 over the natural log of 4 times U of x. And of course, that whole thing times U prime of x. And so, and I'm doing more steps just hopefully so it's clearer what I'm doing here. So this is 1 over the natural log of 4 times U of x is x^2 + x, so x^2 + x.

And we're going to multiply that times U prime of x, so times 2x + 1. So we can just rewrite this as (2x + 1) / (natural log of 4) * (x^2 + x). And we could distribute this natural log of four if we found that interesting, but we have just found the derivative of y with respect to x.

More Articles

View All
An Urgent Warning For Investors | The Coming Recession
What’s up guys, it’s Graham here. So, I think it’s about time that we address a topic that I’m sure a lot of us have considered, and that would be an upcoming recession. After all, in the last few weeks, the yield curve began to flatten as an early recess…
TIL: Hummingbirds Are the World's Hungriest Birds | Today I Learned
If you were to use energy as quickly as a hummingbird, you’d have to eat a fridge full of food or about 300 hamburgers every day in order to survive. They use energy so quickly as they fly, so, so fast. A lot of the flowers they feed on are really delicat…
Consciousness: The Fundamental Reality
Consciousness. It’s our awareness, our understanding, our ignorance. Our daily consciousness leaves out more than it takes in, and due to this, it leaves out important things—things that would help relieve us if we knew them. If we had a higher awareness…
Justification using first derivative | AP Calculus AB | Khan Academy
The differentiable function f and its derivative f prime are graphed. So let’s see. We see the graph of y is equal to f of x here in blue, and then f prime we see in this brownish orangish color right over here. What is an appropriate calculus based justi…
How To Get Rich
world won’t get there by making a social media platform. You aren’t Mark Zuckerberg. The reason these men got to where they are today is because they took a path that no one else ventured down. They made really stupid decisions that led to better decision…
Get a Tour of the Student Experience on Khan Academy
Hi everyone, this is Jeremy Shifling of Khan Academy. I’m joined by our amazing leader of professional learning, Megan Patani. Megan has a real treat in store for you today because she’s going to walk you through not the educator experience that you’re us…