In Exercises 7–38, find the derivative of y with respect to x,...

Question

In Exercises 7–38, find the derivative of y with respect to x, t, or θ, as appropriate.

35. y = ln((x²+1)^5/√(1-x))

Accepted Answer

Welcome back, everyone. Find D Y divided by DX4 Y equals LN of X + 2 to the power of three halves divided by square root of X2 + 4. For this problem, let's begin by applying the properties of logarithms to simplify this function for differentiation. So what we can do is simply use the quotient rule. We're going to begin with LN of X + 2 to the power of 3 halves, and we're going to subtract the bottom function, right? So we're going to take LN of square root of X2 + 4, which can be rewritten as LN X2 + 4 to the power of 1/2. And I notice that from here we can apply the power rules so that we're going to get 3 halves, a line of X + 2. Minus 1/2 a of X2 + 4 and now we can differentiate. So our derivative the Y divided by D X is going to consist of we halves multiplied by the derivative of LNX + 2 minus 1/2 multiplied by the derivative of LN X2 + 4. So here we are factoring out each constant, and now we can focus on each derivative starting with LN of X + 2. It is a composition of two functions. So, first of all, we're differentiating the natural log with respect to X + 2, which gives us 1 divided by X + 2. And according to the chain rule, we're multiplying by the inner function. The derivative of X + 2 is 1, so we're just getting a 1 divided by X + 2. And similarly, we will differentiate a line of X2 + 4. In this case, we're getting 1 divided by X2 + 4 multiplied by x2 + 4 derivative. In this case, the derivative of X 2 is 2 X. The derivative of 4 is 0, so we're going to get 2 X divided by X2 + 4. You know that we have our derivatives, we can substitute back to show that D Y divided by DX is equal to 3 halves. Multiplied by 1 divided by X + 2 minus 1/2 multiplied by 2 x divided by X2 + 4. So we are nearly done. All that we want to do is simplify. We can cancel out -12, and we can basically replace -12 multiplied by 2 with -1. So we can just cancel out 2 in the denominator, right? And now let's go ahead and use the common denominator to write our answer as a single fraction. So we're going to have 2 multiplied by X + 2 multiplied by X2 + 4, right? Now, regarding the first term, we're going to multiply 3 by 1. Notice that we're multiplying 3 by 1. And in this case, our denominator is 2 multiplied by X + 2, so all that we have to do is multiply by X2 + 4, and we're going to subtract X multiplied by. 2 multiplied by X plus 2, right? And now let's go ahead and simplify, so we're going to get 3 X2 + 12 distributing, right? And now we are subtracting X, which is multiplied by 2 X + 4. That is equal to 3 X2 + 12 minus 2 X2 minus 4 X and combining the like terms, that's X2. Minus 4 X + 12. So we can now rewrite our numerator as X2 minus 4 X. 12 and that will be our final answer. Thank you for watching.

In Exercises 7–38, find the derivative of y with respect to x, t, or θ, as appropriate.35. y = ln((x²+1)^5/√(1-x))

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In Exercises 7–38, find the derivative of y with respect to x, t, or θ, as appropriate.
35. y = ln((x²+1)^5/√(1-x))