In Exercises 57–70, use logarithmic differentiation to find th...

Question

In Exercises 57–70, use logarithmic differentiation to find the derivative of y with respect to the given independent variable.

64. y = 1/(t(t+1)(t+2))

Accepted Answer

Welcome back everyone. In this problem we want to use logarithmic differentiation to find the derivative of y with respect to x, for Y equals 1 divided by x multiplied by x + 2 multiplied by x + 3. Now how can we use logarithmic differentiation to help us solve? Well, first, let's try to introduce some logarithms into our problem, right? So let's write downy and take the natural log off both sides. So when we do that, Then the natural log of y will be equal to the natural log of 1 divided by x multiplied by x + 2 multiplied by x + 3, or what we can write as x -1 multiplied by x + 2 to -1 multiplied by x + 3 to -1. No, in this case we can use logarithmic properties to split the right hand side. In other words, If we remember that the log of ABC is equal to the log of A. Plus, let me write that properly here, the log of B plus the log of C, OK, then applying that property. We'll get the natural log of Y to be equal to LNX -1 plus LNX + +21 plus LNX + 3-1. And if we use the other property of logs that says that the log of a to the power of b equals b multiplied by the log of a, then we could also rewrite this then as that is on our right hand side as negative ln x minus LNX + 2 minus LNX + 3. Now we can differentiate implicitly with respect to x. If we take our left hand side, the derivative of Ly will be equal to the derivative of y divided by y, and on our right hand side, the derivative of Lnx with respect to x will be equal to -1 divided by x. For LNX + 2, it will be 1 divided by x + 2. And for LNX + 3, it will be 1 divided by x + 3. OK. So what can we do here? Well, in this case we've almost found the derivative of y with respect to x, but now we need to solve for it. So if we multiply both sides by y, then we get the derivative of y with respect to x or y to be equal to y multiplied by the terms that we have here on our right hand side. OK And remember from our problem statement we already know what Y is. So if we put that into our problem, it's going to be 1 divided by x multiplied by x + 2 multiplied by x + 3, multiplied by the expression that we have from our previous line. So where do we go from here? Well, you know, to simplify, we just need to combine the three terms in parentheses over a common denominator, and notice here the common denominator would be x multiplied by x + 2 multiplied by x + 3. So if we take our expression, let me just, uh, basically copy and paste it into our section below, right. OK, then we know that we could rewrite this as X + 2 Multiplied by x + 3. Plus x multiplied by x + 3 plus x multiplied by x + 2 all dividing. By x multiplied by x + 2 multiplied by x + 3. And now if we simplify our numerator here, we'll have x2 + 5 x + 6 for our first term, x2 + 3 x for our second term, and x2 + 2 x for a second term, and simplifying that even further, you should find that we end up in our numerator with 3 x 2 plus 10 x + 6. So now if we bring this simplified expression back to our original problem. That means the derivative of Y or Y is going to be equal to. 1 divided by x multiplied by x + 2 multiplied by x + 3 multiplied by our expression 3 x 2. Sorry, -3 X2, my apologies. This should be negative here, so that's multiplied by -3 x2 + 10 x + 6 divided by x multiplied by x + 2 multiplied by x + 3. OK, so all of this term before should have been negative, OK, in our numerator. And when we simplify this, no, we get it to be 3 x2 + 10 x + 6 divided by x multiplied by X + 2 multiplied by X + 3. All squared. Thus, this is the derivative of Y, using logarithmic differentiation. Thanks a lot for watching everyone. I hope this video helped.

In Exercises 57–70, use logarithmic differentiation to find the derivative of y with respect to the given independent variable.64. y = 1/(t(t+1)(t+2))

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In Exercises 57–70, use logarithmic differentiation to find the derivative of y with respect to the given independent variable.
64. y = 1/(t(t+1)(t+2))