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.

15. y = sin⁻¹√(1-u²), 0

Accepted Answer

Welcome back everyone. Find dyy divided by dx if y equals inverse of sin of square root of 1 minus x to the power of 44 x between 0 and 1 noninclusive. For this problem, let's recognize that we have a composite function. So what we want to do is simply apply the chain rule. Dy divided by dx is going to be the derivative of the outer function. So the derivative of inverse of sine of square root of 1 minus x to the power of 4. With respect to the inner function, so that would be square root of. 1 minus x to the power of 4. And according to the chain rule, we want to multiply by the derivative of the inner function. So the derivative of the square root of 1 minus x to the power of 4. With respect to X. Well then, so now let's remember the derivative of inverse of sin of x. It is equal to one divided by the square root of. 1 minus x 2. So we can now show that the derivative. Of y with respect to x begins with One divided by. Square root of. 1 minus. We're going to take our inner function, which is square root of 1 minus x 4, and we are going to square it, right? Well done. And now we are multiplying by the derivative of 1 minus x the power of 4, all raised to the power of 1/2. This is how we can replace the radical. And we want to take the derivative. So again, we have a composite function, so we want to apply the chain rule once again. Let's rewrite what we have and simplify. That's 1 divided by square root of 1 minus 1 plus x the power of 4. So this reduces into the square root of x the power of 4. And we are multiplying by Applying the chain rule, the outer function is our exponent, so we want to apply the power rule, which gives us 1/2 multiplied by 1 minus x the power of 4 to the power of 1/2 minus 1, that's -1/22. And based on the chain rule we are multiplying by the derivative of the inner function 1 minus x to the power of 4. Now we want to find the remaining derivative and simplify. Let's begin by simplifying the first term, that's 1 divided by a square root of x to the power of 4, and specifically, that's the absolute value of x 2 or basically x 2, right, because it's always non-negative. And specifically, it must be always positive since division by 0 is undefined and additionally x is between 0 and 1. So we basically get. 1 divided by x2 multiplied by 1. Divided by. To square root of 1 minus x to the power of 4, considering our next term, since we have an exponent of 1/2, we can basically rewrite it in terms of the radical in the denominator. And our final derivative would be 0. The derivative of a constant is 0. Minus the derivative of x to the power of 4 is 4 x cubed using the. Power rule. Well done. And now let's go ahead and simplify x cubed divided by x2 gives us x. And now 4 divided by 2 gives us 2. We can factor out the negative sign, and we are going to get negative. In the numerator That's going to be. To Followed by X. And in the denominator we have square root of 1 minus x to the power of 4, and this is our final answer for this problem. Thank you for watching.

In Exercises 1–24, find the derivative of y with respect to the appropriate variable.
15. y = sin⁻¹√(1-u²), 0<u<1

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In Exercises 1–24, find the derivative of y with respect to the appropriate variable.15. y = sin⁻¹√(1-u²), 0<u<1

Watch next

In Exercises 1–24, find the derivative of y with respect to the appropriate variable.
15. y = sin⁻¹√(1-u²), 0<u<1