Video: MATH-DIFF-INT-2018-S1-Q13

MATH-DIFF-INT-2018-S1-Q13

02:20

Video Transcript

If π‘Ž to the power of 𝑦 is equal to 𝑏 to the power of π‘₯ such that π‘Ž and 𝑏 are in the real positive numbers and π‘Ž is not equal to 𝑏, find 𝑑𝑦 by 𝑑π‘₯.

In order to find 𝑑𝑦 by 𝑑π‘₯, the first thing we’re going to need to do is to get this equation in terms of 𝑦. In our equation, 𝑦 is part of an exponential, as it is in π‘Ž to the power of 𝑦. In order to get the 𝑦 out of the exponential, we’re going to need to perform the inverse operation.

Since we have π‘Ž to the power of 𝑦, we’re going to need to take the log of both sides with a base of π‘Ž. So taking the log to base π‘Ž of both sides of this equation gives us log to base π‘Ž of π‘Ž to the power of 𝑦 is equal to log to base π‘Ž of 𝑏 to the power of π‘₯.

Our next step will be to use a logarithm rule. This logarithm rule tells us log to base π‘˜ of 𝑀 to the power of 𝑛 is equal to 𝑛 timesed by the log to base π‘˜ of 𝑀. What this is telling us is that we can take our power and bring it out to the front of the logarithm. And we can do this to π‘₯ also. This gives us that 𝑦 timesed by log of base π‘Ž of π‘Ž is equal to π‘₯ timesed by log of base π‘Ž of 𝑏.

Next, we’ll be using another logarithm rule. This rule tells us that log to base π‘˜ of π‘˜ is equal to one. And so, therefore, log to base π‘Ž of π‘Ž is also equal to one. This gives us that 𝑦 is equal to π‘₯ multiplied by the log of base π‘Ž of 𝑏. Now we have found 𝑦 in terms of π‘₯. And we’re ready to differentiate to find 𝑑𝑦 by 𝑑π‘₯.

We simply differentiate this whole equation with respect to π‘₯. And this gives us 𝑑𝑦 by 𝑑π‘₯ is equal to log to base π‘Ž of 𝑏. The reason why we get log to base π‘Ž of 𝑏 here is because log to base π‘Ž of 𝑏 is a constant. And π‘₯ simply differentiates to one. So we end up with one timesed by the log to base π‘Ž of 𝑏, leaving us with just the log to base π‘Ž of 𝑏. So this is the solution to the question.

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