In Exercises 59–86, find the derivative of y with respect to t...

Question

In Exercises 59–86, find the derivative of y with respect to the given independent variable.

79. y = θ sin(log₇ θ)

Accepted Answer

Welcome back, everyone. Let Y be equal to T multiplied by a sign of log of base 10 C. Compute Dy divided by DC. For this problem, let's understand that why is. A function that consists of a product of two functions. So to find the derivative of Y, we're going to apply the product rule. According to the product rule, we take our first function, which is T, we're going to take as derivative, multiply by the second function, which is sign of log. Of base 10 of T, and we're going to add T multiplied by the derivative of the second function. So the derivative of sine. Of log 10 of tea. So now, based on this expression, we want to identify two derivatives. The first one is T derivative, which is 1, because that's the independent variable. And for the remaining derivative. Well, it is a composite function, so we want to apply the chain rule. First of all, we're differentiating sine with respect to the inner function and the derivative of sinus cosine, and we're getting cosine of log of base 10 of T. And then the chain rule says that we want to multiply by the derivative of the inner function, so we take the derivative of log of base 10 of T. Remember that the derivative of log of base A of X. is equal to 1 divided by X. Which is multiplied by LN of A. So, in this case, log of base 10 of T derivative is going to be 1. Divided by T, which is multiplied by LN of 0. Meaning The derivative of sine in this case is cosine of log of base 10 of T. Divided by TLN of 10. Well, then we have the two required derivatives, and we can substitute them into the Original equation, so D Y divided by DT is going to be 1 multiplied by. Sign Of log of base 10 of T plus. See multiplied by the derivative of sine. Well, we already know that it's Cosine of log of base 10 of T. Divided by See Ellen of sun. Now we can cancel out T on both sides, and we can use the common denominator to get a neatly looking answer, right? So the common denominator is LN of 10. Sine is going to be multiplied by that value, so we're going to get LN of 10. Multiplied by sign. Of lack of base 10 of tea. And we're going to add. Cosine Of log of base 10 of T, right? So we're adding cosine of log of base 10 of T. And that's going to be our final answer for this problem. Let's label it and thank you for watching.

In Exercises 59–86, find the derivative of y with respect to the given independent variable.79. y = θ sin(log₇ θ)

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In Exercises 59–86, find the derivative of y with respect to the given independent variable.
79. y = θ sin(log₇ θ)