In Exercises 57–70, use logarithmic differentiation to find th...

Question

In Exercises 57–70, use logarithmic differentiation to find the derivative of y with respect to the given independent variable.

66. y = θsin(θ)/√(sec(θ))

Accepted Answer

Welcome back everyone. In this problem we want to use logarithmic differentiation to find the derivative of y with respect to x, for y equals x2 cos x divided by the square root of cosse x. Now, before we can use differentiation, let's first try to express our equation in terms of logarithms, OK, so we know that why. Equals x2 cus x divided by the square root of cos x. And we could rewrite this as x2 + x multiplied by cosse x to the power of -5. And that will be pretty helpful because then if we take the natural logs of both sides, then we'll get the natural log of Y to be equal to the natural log of our expression on the. Right hand side of our equation. Now where can we go from here? Well, recall one properties of logs tells us that if we're finding the log of a product, for example, ABC, we could rewrite it as log A plus log B plus log C, OK. So if we apply that property to our expression here, then we could rewrite our right hand side as the natural log of x2 plus LN + x plus LN cosec x to the power of -522. And now, if we apply another property of logs to simplify this even further, remember that if we're finding the log of A to the power of B, we could rewrite it as blog A. So if we use that here, then we could rewrite it further as 2 LNX plus LN X, OK. Minus 12 multiplied by the LN of course x, so we bring down the powers of those expressions that had powers. Now that we have that, let's see if we can differentiate both sides with respect to X, OK. So differentiating. Both sides. With respect to eggs, OK. Then the derivative of LNY, we should get it to be 1 divided by y multiplied by DYDX. On the other hand, the derivative of LNX is 1 divided by x. The derivative of Ellen cost X is negative 10 x. And the derivative of Ellen Csek x is negative cut x. Now, if we simplify this further, we'll get 2 divided by x minus 10 x plus 12 of cut x. Now where do we go from here? Well, now that we have DYD X, we can solve. If we multiply both sides by y, then we get DYD X to be equal to y multiplied by this new expression on our right hand side. And now remember at the start of our problem we know ye equals x2 x divided by the square root of cosse x. So now to express our derivative purely in terms of x, it's going to be x2 cost x divided by the square root of cos x multiplied by 2 divided by x minus 10 x plus a half of x. This is the derivative of Y with respect to X using logarithmic differentiation. Thanks a lot for watching everyone. I hope this video helped.

In Exercises 57–70, use logarithmic differentiation to find the derivative of y with respect to the given independent variable.66. y = θsin(θ)/√(sec(θ))

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In Exercises 57–70, use logarithmic differentiation to find the derivative of y with respect to the given independent variable.
66. y = θsin(θ)/√(sec(θ))