Calculate the derivative of the following functions (i) using ...

Question

Calculate the derivative of the following functions (i) using the fact that b^x = e^{xIn b} and (ii) using logarithmic differentiation. Verify that both answers are the same.

y = (4x+1)^{In x}

Accepted Answer

Hi everyone. Let's take a look at this practice problem. This problem says to find the derivative of the given function by using logarithmic differentiation or by using the property B rated to the power X is equal to E rate to the quantity of X multiplied by the natural log of B. And we're given the function that Y is equal to the quantity of 13 x plus 2 in quantity raised to the natural log of X. So we're asked to find a derivative of our function Y, and here we're actually going to choose to use logarithmic differentiation. So to use logarithmic differentiation, we need to first take the natural log of our function y. So we'll have the natural log of Y is equal to the natural log of the quantity of 13 x plus 2 in quantity raised to the natural log of X. So we're going to rewrite the right-hand side using um our properties of logarithms, mainly our power rule. So we'll have the natural log of Y is equal to the natural log of X multiplied by the natural log of the quantity of 13 X plus 2 in quantity. So now what I want to do is take a derivative with respect to X of both sides of our equation. So on the left hand side, I will have the derivative with respect to X of the natural log of Y. So recall that when you take a derivative of your natural log function, you will get 1 divided by the argument, which in this case is 1 divided by Y. But we'll have to also apply our chain rule here, so we'll need to multiply this by the derivative of our argument, which in this case is going to be the derivative of Y, with respect to X. And then this is going to be equal to. On the right hand side, I'm going to have to apply our product rule. So recall for your product rule that you're going to have the derivative of the first term of our product, which in this case is going to be the natural log of X. You want to take the derivative with respect to X of the natural log of X, I will get 1 divided by X. And then that gets multiplied by our second term, which is going to be the natural log of the quantity of 13 x plus 2 in quantity. Then we'll have plus the first term in our product, which is the natural log of X, multiplied by the derivative of our second term. So we'll have the derivative with respect to X of the natural log of the quantity of 13 x plus 2. So again, here we're taking the derivative of our natural log function. And so that will give me 1 divided by the quantity of 13 x plus 2, since that is my argument in quantity, and then that gets multiplied by the derivative of our argument, again applying our chain rule. So the derivative with respect to X of the quantity of 13 x plus 2 is 13, we multiply our quantity here by 13. So what I want to do is to find the derivative of our functions, so we need to solve this for the um derivative of Y with respect to X. So to do that, we'll multiply both sides by Y. So we'll have the derivative of Y with respect to X is equal to Y multiplied by the quantity, and here we'll just simplify our remaining terms. So we'll have the natural log of the quantity of 13 X plus 2 in quantity, divided by X. Plus 13 multiplied by the natural log of X divided by the quantity of 13 x plus 2 in quantity. So now we just need to substitute back in our original expression for Y. and so we'll have the derivative of Y with respect to X. Equal to the quantity of 13 x plus 2 in quantity raised to the natural log of X. Multiplied by the quantity of the natural log of the quantity of 13 X. Plus 2 in quantity, divided by X. Plus, 13, multiplied by the natural log of X. Divided by the quantity of 13 X plus 2 in quantity. So this here is our final answer for this problem, since we have solved for the derivative of Y with respect to X, which is what the question was asking us to do. So I hope that this explanation has been useful, and I'll see you in the next video.

Calculate the derivative of the following functions (i) using the fact that b^x = e^{xIn b} and (ii) using logarithmic differentiation. Verify that both answers are the same.
y = (4x+1)^{In x}

Key Concepts

Derivative

Exponential and Logarithmic Functions

Logarithmic Differentiation

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