In that case, F prime of X is going to be the derivative of this whole thing with respect to X plus five over X minus one which is going to be one over X plus five over X minus one times the derivative, times the derivative with respect to X of X plus five over X minus one. Simplify this expression using this property and just to try and power through this using the chain rule. Now what's the hard way you might be thinking? Or maybe you did do it when you tried to approach it on your own. And the derivative of this, well, let's see, we're going to have a minus sign there and the derivative of the natural log of X minus one with respect to X minus one is going to be one over X minus one and the derivative of X minus one with the respect to X is just one you just multiply this by one, it doesn't really change the value. I'm just applying the chain rule here, and that's just going to be one. And when we take the derivative now with respect to X, F prime of X, well this is going to be the derivative of the natural log of X plus five with respect to X plus five, so that's going to be one over X plus five times the derivative of X plus five with respect to X. We can write F of X as being equal to the natural log of X plus five minus the natural log of X minus one. It from a point of view in terms of having to So if we just apply this property right over here, and just simplify this expression, or at least simplify So this is just going to be equal to the natural log of A minus the natural log of B. The easy way is to recognize your logarithm properties, to remember that the natural log of A over B. And I encourage you to pause this video and try to figure it out on your own. And what we want to figure out is what is F prime of X. That derivative approaches 0, that is, becomes smaller.ī) when x is less than 1 and becomes smaller.Voiceover:Let's say that we've got the function F of X and it is equal to the natural log of X plus five over X minus one. Calculate the derivative of lnĪccording to the rule for changing from base e to a different base a:Ī) when x is greater than 1 and becomes larger. When y = e u( x), then according to the chain rule:Įxample 4. The derivative of e with a functional exponent In the system of natural logarithms, in which e is the base, we have the simplest constant possible, namely 1. ( Lesson 39 of Algebra.) When we calculate that derivative below, we will see that that constant becomes ln a. Where k is the constant of proportionality. Therefore, to say that the rate of growth is proportional to its size, is to say that the derivative of a x is proportional to a x. The more individuals there are, the more births there will be, and hence the greater the rate of change of the population - the number of births in each year.Īll exponential functions have the form a x, where a is the base. The bigger it is at any given time, the faster it's growing at that time.
For we say that a quantity grows "exponentially" when it grows at a rate that is proportional to its size. What does that imply? It implies the meaning of exponential growth.