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.

29. y = ln(1/(x√(x+1)))

Accepted Answer

Welcome back, everyone. Find the derivative of Y with respect to X of Y equals Ln of square root of X + 2 divided by X2, where X is greater than -2. For this problem, whenever we are differentiating, we want to simplify functions, right? And because we have natural log of a ratio, let's apply the quotient rule of logarithms. We can show that Y is equal to LN of square root of X + 2 minus LNX2 because we have a ratio, we can essentially distribute and subtract two logarithms. A land of the numerator minus a land of the denominator and now additionally we can apply the power rule because a land of square root of x + 2 can be rewritten as LN of x + 2 raise the power of 1/2. So now applying the power rule we are going to bring down 1/2. And we're going to get 1/2 LN of X + 2. Minus applying the same power rule for the next term, we get 2 LN of X. Well then, so now we have a simpler form, so we are ready to differentiate Y, which is the Y divided by D X. Is going to be equal to, we can factor out the constant 1/2 multiplied by the derivative of LNX + 2 minus 2 multiplied by the derivative of LN of X. And now let's recall that the derivative of LN of X is one divided by X. And to find the derivative of LNX + 2, we're going to apply the chain rule since we have a composite function. According to the chain rule, we take the derivative of the natural log with respect to X + 2, which is 1 divided by X + 2. And we're multiplying by the derivative of the inner function, which is X + 2. The derivative of X + 2 is equal to 1, because the derivative of X is 1 and derivative of 2 is 0. So we simply get 1 divided by X + 2. And now let's substitute these expressions into Y. We're going to get 1/2 multiplied by 1. Divided by X + 2 minus. 2 multiplied by 1 divided by X. All that we have to do is simplify. So this is equal to. One divided by. Our denominator can be written in a factored form, so we're going to write one divided by. 2 Multiplied by x + 2, and we're subtracting 2 divided by X. And from here, let's use the common denominator to write the answer using one fraction. So, the first term will be multiplied by X. And for the second one, we are multiplying the numerator by 2, which is multiplied by X + 2, so we're subtracting 2 multiplied by 2. X + 2. And we're dividing by the common denominator, which is 2 multiplied by X, multiplied by X + 2. You know, let's simplify the numerator, we get X minus. Or X. Minus or multiplied by 2, that would be -8, right? And we are dividing by 2 X X plus 2. Combining the like terms, we get X minus 4 X, that's -3 X. So we get -3 X minus 8 in the numerator. We can also factor out the negative sign in front of the fraction, so both terms. Get positive signs in the numerator because they're saying negative sign in front of the fraction. And therefore, the final answer is negative. In the numerator, we have 3 X + 8, and the denominator we have 2 X multiplied by X + 2. Thank you for watching.

In Exercises 7–38, find the derivative of y with respect to x, t, or θ, as appropriate.29. y = ln(1/(x√(x+1)))

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