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Problem 25 Easy Difficulty

Use Formula 11 to find the curvature.
$$y=x e^{x}$$

Answer

$$
\frac{e^{x}|x+2|}{\left[1+\left(x e^{x}+e^{x}\right)^{2}\right]^{\frac{3}{2}}}
$$

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Video Transcript

in this problem, we want to find the curvature Kappa of the function. Why equals X times e to the power of X? You know that we're gonna need the first and second derivative. So let's go ahead and take first derivative y prime. And using the product rule, we end up getting X times each the X plus he to the X and then taking the second derivative, my double prime. We end up getting X times e to the power of X plus e to the power of X. That's the product rule in our first term. And then if we take the derivative of the second term, we get e to the power of X, which we can combine our like terms and we end up with X times e to the power of X plus two e to the power of X, which we can further simplify if you want, by factoring or even ex out, which would give us explodes two times you the power of X. We can then use those components to find Kapo, the curvature cap of X in the numerator. We're going to have the absolute value of our second derivative so X plus two times e power of X and in our denominator will have one plus our first derivative exceeded the power of X plus Eat of power of X all squared. And then that entire denominator is going to be raised to the three house power and we can simplify things a little bit further. We know that E to the power of X is always going to be positive so we can pull it out of the square or out of the absolute value sign. So we have each the power of X Times X plus two, then in our denominator and we can leave that the same so one plus x times each of the power of X plus eat of power of X all squared raised to the three house power. And that is the curvature of the function. Why equals X times eat of power of X

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