![]() ![]() Differentiate the outer function, keeping the inner function the same.How to Do the Chain Rule To do the chain rule: Such functions must be differentiable themselves.The function must be a composite function of two or more functions.To use the chain rule, the following rules are required: In words, the chain rule requires finding the derivative of the outer function while keeping the inner function the same and then multiplying this by the derivative of the inner function. g(□) is the inner function and f(□) is the outer function. The chain rule is defined as, where u is a function of □ ( u = g(x) ) and y is a function of u ( y = f(u) ).Īlternatively, the chain rule can be written in function notation as F'(□) = f'(g(□)).g'(□), where F(□) = f(g(□)). Times three to the x power.The chain rule is used when a function is within another function. So it's equal to eight natural log of three times three to the x. Natural log of our base, natural log of three times three to the x. Of this right over here is going to be, based on what we just saw, it's going to be the So if I want to find the derivative with respect to x of eight times three to the x power, well what's that going to be? Well that's just going to be eight times and then the derivative And so we can now use this result to actually take theĭerivatives of these types of expressions with bases other than e. If you're taking theĭerivative of a to the x, it's just going to be the natural So if you're taking theĭerivative of e to the x, it's just going to be e to the x. It all simplifies to the natural log of a times a to the x, which is a pretty neat result. And so this is going to give us the natural log of a times e to the natural log of a. Of natural log a times x, it's just going to be natural log of a. So that's just going to be, so times the derivative. ![]() Well natural log of a, it might not immediately jump out to you, but that's just going to be a number. Of that inside function with respect to x. And so, this is going to be equal to e to the natural log of a times x. So e to the natural log of a times x with respect to the inside function, with respect to natural log of a times x. So what we will do is we will first take the derivative ![]() And now we can use the chain rule to evaluate this derivative. So that's going to be the same thing as e to the natural log of a, natural log of a times x power. Raising our original base to the product of those exponents. If I raise something to an exponent and then raise that to an exponent, that's the same thing as Our exponent properties, this is going to beĮqual to the derivative with respect to x of, and I'll just keep color-coding it. We're going to raise that to the x power. If a is the same thing asĮ to the natural log of a, well then this is going to be, then this is going to beĮqual to the derivative with respect to x of e to the natural log, I keep writing la (laughs), to the natural log of a and then we're going to Don't just accept this as a leap of faith. ![]() So if you actually raise e to that power, if you raise e to the power you need to raise e too to get to a. What is the natural log of a? The natural log of a is the power you need to raise e to, to get to a. Now if this isn't obvious to you, I really want you to think about it. Well all right, a as being equal to e to the natural log of a. Something with e as a base? Well, you could view a, you could view a as being equal to e. Well can we somehow useĪ little bit of algebra and exponent properties to rewrite this so it does look like Knowledge that the derivative of e to the x, is e to the x. Where a could be any number? Is there some way to figure this out? And maybe using our What is the derivative, what is the derivative with respect to x when we have a to the x, The slope at any point, is equal to the value Though when you have an exponential with your base right over here as e, the derivative of it, So we've already seen that the derivative with respect to x of e to the x is equal to e to x, which is a pretty amazing thing. Want to do in this video is explore taking the derivatives ![]()
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |