In Exercises 25–36, find the derivative of y with respect to t...

Question

In Exercises 25–36, find the derivative of y with respect to the appropriate variable.

33. y = csch⁻¹(1/2)^θ

Accepted Answer

Welcome back everyone. Find the derivative of y with respect to theta if y equals inverse of cosse of 1/3 raise to the power of theta. So for this problem we want to find the derivative of y and y is a composite function. So what we want to do is simply apply the chain rule of differentiation, right? Let's begin with the fact that the derivative of inverse of cosesh. Of x is equal to -1. Divided by the absolute value of x, which is multiplied by the square root of 1 + x 2. So in this case x is changed into a function in terms of theta. Let's go ahead and apply the chain rule so that y is going to be equal to -1 divided by x becomes 1/3 to the power of theta. So we get -1 divided by the absolute value of 1/3 erase the power of theta. Followed by a square root of one plus. 1/3 to the power of theta. Squared, right? And after that we want to apply the chain rule, so we are multiplying by the derivative of the inner function which is the derivative of 1/3. Raised to the power of theta. Let's go ahead and simplify. So first of all, we will get -1 divided by, we can drop the absolute value since we have 1/3 to the power of theta. This is an exponential function. The leading coefficient is 1, so it's always positive. So we get -1 divided by 1/3. To the power of theta. Which is multiplied by 1 plus applying the properties of exponents that would be 1/3. Raise the power of 2 multiplied by theta or 2 theta. Now let's find the remaining derivative. We have 1/3 to the power of theta derivative, and let's remember that this is an elementary derivative that would be 1/3 raise to the power of theta multiplied by ln of 1/3, right? And now LN of 1/3 can be written as LN 3 using the properties of logarithms. So we can just write multiplied by ln of 3. Well then, now let's notice that we can cancel out 1/3 to the power of theta in the denominator and the numerator, and we can also cancel out the two negative signs. We can now write Ean of 3. In the numerator And in the denominator we have square root of. 1 + 1/3 raise to the power of 2 theta. And that's our final derivative for this problem. Thank you for watching

In Exercises 25–36, find the derivative of y with respect to the appropriate variable.33. y = csch⁻¹(1/2)^θ

Watch next

In Exercises 25–36, find the derivative of y with respect to the appropriate variable.
33. y = csch⁻¹(1/2)^θ