{Use of Tech} A family of superexponential functions Let ƒ(x) ...

Question

{Use of Tech} A family of superexponential functions Let ƒ(x) = (a + x)ˣ , where a > 0.

c. Compute ƒ'. Then graphƒ and ƒ' for a = 0.5, 1, 2, and 3.

Accepted Answer

Hello there. Today we're going to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Let H of X equal 2 x + 1 to the power of X, where X is a real number. Determine the derivative of H and using a graphing utility plot H. Fantastic. So it appears for this particular prompt we're asked to solve for two separate answers. Our first answer is we're asked to find this derivative of this function of H X. That's our first answer. And once we've found the derivative of H of X, we're asked to plot our new graph for H X. So we're trying to plot the graph of our differentiation when we perform the derivative of H of X, and we're plotting the curve for the derivative. So now that we know what we're solving for, our first step is we need to determine the derivative of, and once again, let us know we're trying to find the derivative of H of X is equal to 2 X plus 1 to the power of X. That is the function that we're trying to take the derivative. Of, so we're trying to find H X. So in order to find H X, which once again is the derivative of our function of H of X, we need to use logamithic differentiation. So we need to recall and use logamithic differentiation. So with that in mind, we need to take the natural log first, so we need to take the natural log of H of X, so the natural log of H of X. And the natural log of H of X is going to be equal to X multiplied by the natural log of 2 X plus 1, fantastic. So now we need to do. Differentiation implicitly. So we need to differentiate implicitly. So using implicit differentiation, we need to take H of X, which once again, this prime symbol denotes the first derivative of H of X. So we need to take H X divided by H of X, and say that it's equal to the natural log of 2 X plus 1, and we need to add. 2 X divided by 2 X plus 1. Fantastic. We are on a roll. So now our next step is we need to solve for H X, we need to perform the differentiation. So when we perform the differentiation, we will find that H of X and let us write it down below. So our derivative of H X H X when we perform the differentiation is gonna be equal to 2 X + 1 to the power of X multiplied by the natural log of 2 X + 1. Plus 2 x divided by 2 X plus 1. Fantastic. And then don't forget your parenthesis. And that's it. That is our derivative for H of X. So HX once again has to be equal to 2 x + 1 to the power of X multiplied by the natural log of 2 X + 1. Plus 2 x divided by 2 X plus 1. That is our derivative for our function H of X. So that is one of our answers. So now when we take this derivative, so when we take this function and we plug it into our graphing software of choice, we should produce the following graph. So you could use MATLAB or you can use whatever software makes the most sense to you. But once you've found the software that makes the most sense to you and you plug it in, you should produce the following curve, which our curve is going to be represented in blue. And as you can see, it starts in the negative 3rd quadrant, works its way up towards what appears to be crossing through the origin point. And then it starts to curve upwards into the positive direction within the first quadrant of our graph. And of course, the Y axis is in the vertical direction and the X-axis is in the horizontal direction. And it appears this graph looks to be cubic in nature. And that's it. We've officially solved for this problem. Hooray. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

{Use of Tech} A family of superexponential functions Let ƒ(x) = (a + x)ˣ , where a > 0.

c. Compute ƒ'. Then graphƒ and ƒ' for a = 0.5, 1, 2, and 3.

Key Concepts

Differentiation

Exponential Functions

Graphing Functions

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