In Exercises 1–24, find the derivative of y with respect to th...

Question

In Exercises 1–24, find the derivative of y with respect to the appropriate variable.

21. y = z arcsec(z) - √(z² - 1), z>1

Accepted Answer

Welcome back everyone. Compute the derivative of y with respect to u, for y equals u multiplied by inverse of sin of u minus square root of 1 minus u 2, where u is between -1 and 1. For this problem we simply want to find the derivative of y, which is Y. Let's notice that one of our terms is a product, so we are going to apply the product rule, that would be you multiplied by Inverse of sin of u derivative minus. The derivative of square root of 1 minus u 2. So let's focus on each derivative, starting with the first one. We are going to apply the product rule u multiplied by inverse of sin of u derivative is going to be equal to u derivative. Multiplied by inverse of sine of u. Plus u multiplied by inverse of sine of u derivative. Now the derivative of u is equal to 1, and we're going to get 1 multiplied by inverse of sin of u or basically inverse of sin of u plus u multiplied by. Inverse of sin of u derivative is equal to 1 divided by square root of 1 minus u 2. So now we can just write you in the numerator. To get a simpler form. Well then, this is how we get the derivative of the product. Now let's focus on the remaining derivative. We are going to rewrite the radical in terms of an exponent. We have 1 minus u2 raise to the power of 1/2. This is what the square root is, and we want to differentiate this function. Here we have to notice that this is a composite function, so we want to apply the chain rule. Starting with the power rule, we get 1/2. Multiplied by 1 minus u 2 raised to the power of 1/2 minus 1, which is -1/2. And according to the chain rule we are multiplying by the derivative of the inner function 1 minus u squared. Now let's go ahead and simplify. This is 1 divided by 2, square root of 1 minus u2 multiplied by the derivative of 1 is 0, and the derivative of u2 is -2 u. Now we can cancel out the two and we're going to get negative you. Divided by square root of 1 minus u 2. And now we can combine the results. Combining the results, we are going to get Y equals. In verse of sign. Of you plus you Divided by a square root of 1 minus u2 minus g u divided by the square root of 1 minus u2, so we're basically adding u. Divided by a square root of 1 minus u 2. We can combine those terms and we can simply write 2 u divided by a square root of 1 minus u 2. Well then, this is how we get our final derivative. Let's label it and thank you for watching.

In Exercises 1–24, find the derivative of y with respect to the appropriate variable.21. y = z arcsec(z) - √(z² - 1), z>1

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In Exercises 1–24, find the derivative of y with respect to the appropriate variable.
21. y = z arcsec(z) - √(z² - 1), z>1