Product Rule for three functions Assume f, g, and h are diff...

Question

Product Rule for three functions Assume f, g, and h are differentiable at x.

b. Use the formula in (a) to find d/dx(e^x(x−1)(x+3))

Accepted Answer

Hello. In this video, we are going to be using the product rule to find the derivative of the given function. The function given to us in this problem is 2E raise to the power of X, multiplied by the quantity 3X minus 2 multiplied by the quantity 2 X plus 1. So, before we jump into this problem, let's quickly review what the product rule states. The product rule states that if we want to take the derivative of the product of two functions, such as F of X multiplied by G of X, the derivative is defined as F of X, multiplied by the derivative of G of X, plus G of X multiplied by the derivative of F of X. So the product rule allows us to take the derivative of the product of two functions. But what's tricky about this problem is that we have a product of 3 functions. So before we use the product rule, let's go ahead and simplify the original function by combining two of the terms in the product. In this case, let's go ahead and combine 3X minus 2 and 2 X + 1 by foiling them together. If we foil 3X minus 2 and 2 X + 1, it will simplify to leave us with 6 X squared minus X minus 2, and this will be multiplied by 2 E raise to the power of X. Now what we have done is we have simplified the original function to be the product of two functions. Now, if we want to take the derivative of this function, we can use the product rule. We will go ahead and define F of X. As 2E to the power of X, and we will define G of X as 6 X squared minus X minus 2. Now what we need to in order to use the product rule is we need the derivatives of F of X and Geovex. By using the exponential rule, the derivative F of X will equal to 2 to the power of X. And by using the power rule, the derivative G of X will equal to 12 X minus 1. Now that we have the 4 functions that we need, we can go ahead and use the product rule. So the product rule will be F of X, which is 2 to the power of X, multiplied by the derivative of G of X, which is 12X minus 1. Plus, the original G of X, which is 6 X squared minus X minus 2, multiplied by the derivative of F of X, which is 2 E to the power of X. Now what we need to do is we need to simplify our derivative. We will start by distributing 2 E to the power of X to 12 X minus 1, and we will do the same by distributing 2 E to the power of X to 6 X2 minus X minus 2. This will simplify the derivative to be 24 E to the power of X, multiplied by X minus 2 E to the power of X, plus 12. E to the power of X. Multiplied by X2d. 2 to the power of X multiplied by X minus 4E to the power of X. Now what we need to do is we need to go ahead and combine like terms. The first thing we can combine will be 24 E to the power of X multiplied by X, and -2 E X minus X. We can also combine -2 E to the power of X with -4 E to the power of X. If we combine these terms together, we will be left with 12 E to the power of X multiplied by X2, plus 22 E to the power of X multiplied by X minus 6 E to the power of X. Now notice that every term in our derivative has a multiple of 2 and a multiple of E to the power of X. We can go ahead and factor out 2 E to the power of X from our derivative, and that will leave us with 2 E to the power of X, multiplied by the quantity, 6 X squared. Plus 11 X minus 3. And this is going to be the simplest form of the derivative of the function. And with that being said, the solution to this problem is going to be B. So, I hope this video helps you in understanding on how to use the product rule, and I'll go ahead and see you all in the next video.

Product Rule for three functions Assume f, g, and h are differentiable at x.
b. Use the formula in (a) to find d/dx(e^x(x−1)(x+3))

Key Concepts

Product Rule

Chain Rule

Exponential Functions

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