Find f′(1) when f(x) = x^(1/x).

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Accepted Answer

Welcome back, everyone. Find the value of H E where H X is equal to 3x to the power of 1 divided by 3X and E is the base of the natural logarithm. What we're going to do is simply express HX as Y. So Y is equal to 3 X to the power of 1 divided by 3 X. And for this problem because we have a function of X raised to an exponent involving X, well the easiest way is to use logarithmic differentiation. So we're going to take the natural logarithm of both sides, and we're going to get LN of Y equals LN 3 X to the power of 1 divided by 3 X. And from here we can apply the properties of logarithms. LNF Y is going to be equal to 1 divided by 3 X. We can bring down the exponent in front of the logarithm. And then we're multiplying by LN of 3 X. And from here we can differentiate implicitly. So on the left hand side we're evaluating the derivative of LN of Y, right? So we're going to see that on the left we're evaluating LN of Y. Prime, and on the right hand side, we want to value the derivative of the product, 1 divided by 3X multiplied by LN of 3 X. So on the left hand side, the derivative of LN of X overall if we remember, is 1 divided by X. So here we're going to get 1 divided by Y, but Y is dependent on X, so we need to apply the chain rule and multiply that by Y. On the right hand side we're going to apply the product rule. So first of all we're taking the derivative of 1 divided by 3 X and we're going to factor out the constant, so that's 1/3 multiplied by the derivative of X to the power of negative 1. So we are rewriting 1 divided by X as X to the power of negative 1 because we want to apply the power rule. This derivative is then multiplied by the second function, so we're going to multiply that by LN of 3 X, and we're going to add the first function, which is 1 divided by 3 X, multiplied by the derivative of the second function, which is the derivative of LN of 3 X. And now on the left hand side we get Y divided by Y equals. Now applying the power rule, we're going to get negative 1/3. Because we bring down -1, we get 1/3, and now X gets a new exponent. -1 minus 1 gives us -2, so we get X at the bottom squared, right, because X to the power of negative is the same thing as 1 divided by X2. This is multiplied by LN of 3 X, and then we are adding 1 divided by 3 X multiplied by the derivative of LN3X based on the derivative of LN X. This is 1 divided by 3 X, but we also need to apply the chain rule once again. And the derivative of 3 X is simply 3, right, so we can cancel out that 3 because we have 3 in the denominator as well. So now Y is equal to, we can multiply both sides by Y. And we're going to get negative LM of 3 X divided by 3 X squad. That's going to be our first denominator plus 1 divided by we x2. We have a common denominator already, and we can say that Y is equal to. Now what is Y? Well, Y is 3 X. The power of 1 divided by 3 X and then we are simply going to write our fraction so. At the bottom, we have V X squad. And now at the top, we have a sum of two terms. Actually, it's the difference we can write 1 minus LN of 3 X. So we have our derivative in its general form, right? But what we want to do is simply evaluate it at a specific point. We want to know what H at point E is, and this means that we want to substitute X equals E. So we're going to get. We because now we're replacing X with E. To the power of 1 divided by 3 e multiplied by 1 minus Ln. Of 3 E. Divided by V X 2, which becomes VE2. What we're going to do from here is just simplify. So H prime. At point E is equal to. First of all, we're going to write. 3E to the power of 1 divided by 3E. Multiplied by 1 minus applying the properties of logarithms, we are going to get minus LN 3 minus LN of E and LN of E is 1, right? So minus LN of 3 minus LN of E divided by wee 2. So once again, H of E is going to be equal to. We eat to the power of 1 divided by 3 E multiplied by. Now let's notice that 1 minus LN of E gives us 0 because that's 1 minus 1, and we can factor out the negative sign, so we can factor the negative sign in front of the fraction, and this gives us -3e the power of 1 divided by 3 E. Multiplied by ln of 3 divided by 3 e squared. There is no necessity to distribute the exponent and then simplify the E part right, as well as try to combine the constants. It's perfectly fine to leave it in this form, and we're going to label it as our final answer. Thank you for watching.

Find f′(1) when f(x) = x^(1/x).

Key Concepts

Derivative

Chain Rule

Logarithmic Differentiation

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