In Exercises 1–24, find the derivative of y with respect to th...

Question

In Exercises 1–24, find the derivative of y with respect to the appropriate variable.

7. y = log₂(x²/2)

Accepted Answer

Welcome back, everyone. Find D Y divided by DX4 Y equals slog of base 5 of 3 X cubed. Before we begin finding the derivative, let's apply the properties of logarithms to obtain an expression that is simpler to differentiate. We're going to have log of base 5 of 3 X cubed. Notice that we have a product between 3 and X cubed. So we can first of all apply the product rule of logarithms. We're going to get a log of base 5 of 3 plus log of base 5 of X cubed. Now, additionally, we can apply the power rule and we can bring down the exponent. To get Y equals log of 5 of 3 plus we log of 5 of X, and now we can differentiate the derivative of Y, we can use a shorthand notation Y, right? It's going to be equal to. The derivative of log of base 5 of 3. Plus 3 multiplied by the derivative of log of base 5 of X we can factor out the constant. Notice that the first term is a constant log of base 5 of 3 does not contain any variable, so it's derivative is 0 because it's a constant, and we can focus on 1 and only derivative, which is a log base 5x derivative. This is an elementary derivative. Its value is a 1. Divided by X, which is multiplied by LN of the base of the logs, so in this case LN of 5, right? And what we can do is now substitute. So we get 3 multiplied by 1 divided by. XLN 5, which is simply equal to we divided by XLN 5, and that's our final answer for this problem. Thank you for watching.

In Exercises 1–24, find the derivative of y with respect to the appropriate variable.7. y = log₂(x²/2)

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In Exercises 1–24, find the derivative of y with respect to the appropriate variable.
7. y = log₂(x²/2)