In Exercises 115–126, use logarithmic differentiation or the m...

Question

In Exercises 115–126, use logarithmic differentiation or the method in Example 6 to find the derivative of y with respect to the given independent variable.

124. x^(sin y) = ln y

Accepted Answer

Welcome back everyone. Use logarithmic differentiation to find Dy divided by dx4x to the power of cosine y equals y2. So for this problem, let's go ahead and rewrite the relationship x to the power of cosiney equals y2. And let's remember that whenever we want to apply logarithmic differentiation, we want to take a line of both sides to begin with. So we're going to take ln off the left hand side, so that's ln of x to the power of cosine y equals ln of the right hand side, so ln of y 2, and from here we can apply the power rule of logarithms so that we get cosiney multiplied by ln of x equals. 2 l and of Y and from here we can take the derivative of each side. So we're going to take the derivative of the left hand side and the derivative of the right hand side. So now on the left hand side, we have a product. So we once applied the product rule that's cosiney derivative multiplied by een of x. Plus cosiney multiplied by eon of x derivative and on the right hand side we have 2 multiplied by ln of y derivative, which is 1 divided by y multiplied by y derivative because y is a function of x we simply want to apply the chain rule, right. And now let's identify each derivative. The derivative of cosine is negative signs. We get negative sign of y. Multiplied by y derivative once again we want to apply the chain rule followed by eon of x plus cosine of y multiplied by 1 divided by x, which is the derivative of ln of x and on the right hand side we get 2 y divided by y. Now, what we want to do is solve for a Y, right? We want to identify the derivative. So what we can do is simply multiply both sides by y and divide it by 2. So we are going to get y divided by 2 in parentheses. We will have all of the left-hand side, but notice that this is going to give us Y on the right hand side, even though we still have Y on the left. So instead, what we can do is Combine the Y terms on the same side. So on the left-hand side we're going to leave cosine of Y divided by x. Cosine of y divided by x, and on the right-hand side we have 2y divided by y. And we are going to add sin y multiplied by ye of x, right? So plus y ln of x sin of y, and from here we can factor out y. So let's go ahead and do that. This is equal to y in parentheses 2 divided by y plus eon of x sin of y. And from here, we can solve for Y because we have all of the terms on the right-hand side with Y. So we're going to take the left hand side, which is cosiney divided by X. And we want to divide by the factor. Adjacent to Y, which is 2 divided by y plus. LN of Xin of Y. What we're going to do is. Simplify, right, because we have Y, which is exactly what we're looking for, but we want to write a simpler form. So what we can do is simplify the common denominator and perform the division. So that would be cosine of y divided by x, that's our numerator multiplied by. In the denominator, the common denominator is going to be Y, right? So when we are dividing, we're going to flip our fractions so that we will have Y in the numerator and in the denominator that will be 2. Plusy multiplied by ln of x sin of y. So, 2. Plusy multiplied by. X I'm sorry, LN of X sign of Y. So now let's check our terms, cosine of y divided by x multiplied by y, right, divided by 2 plus y ln x sin of y. So everything makes sense. And now we can combine to show that this is simply y multiplied by cosine of y. Divided by x, which is multiplied by 2 plus y, ln of x sin of y, and that's our final answer, our final derivative. Thank you for watching.

In Exercises 115–126, use logarithmic differentiation or the method in Example 6 to find the derivative of y with respect to the given independent variable.124. x^(sin y) = ln y

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In Exercises 115–126, use logarithmic differentiation or the method in Example 6 to find the derivative of y with respect to the given independent variable.
124. x^(sin y) = ln y