In Exercises 25–30, use logarithmic differentiation to find th...

Question

In Exercises 25–30, use logarithmic differentiation to find the derivative of y with respect to the appropriate variable.

27. y = (((t+1)(t-1))/((t-2)(t+3)))^5, t>2

Accepted Answer

Welcome back everyone. Use logarithmic differentiation to compute dyy divided by dx4 Y equals x + 2 multiplied by x minus 3, divided by x minus 1 and divided by x + 4, all raised to the power of 4, where x is greater than 3. For this problem we want to use logarithmic differentiation, so let's take LN of both sides. LN of Y is going to be equal to LNX + 2 multiplied by X minus 3. Divided by the product of x minus 1 and x + 4. All raised to the power of 4. Now, what we can do is simply use the properties of logarithms to bring down the exponent, and we can write 4 LN of our function, right? And from here, we're going to apply another property of logarithms, the property of product and quotient. So a lot of y is going to be equal to 4 multiplied by. In parentheses we can split our logarithm into ln x + 2 plus ln of x minus 3 and we're going to subtract the denominator factors. So minus ln of x minus 1 minus ln of x + 4, and from here we can proceed with differentiation on the left hand side ln of y derivative. We went to calculate the derivative of all of y, right, and on the right-hand side, we went to calculate the derivative of 4 multiplied by the sum and differences of logs. So on the left-hand side, the derivative of ln of y is 1 divided by y multiplied by y derivative because y is a function of x. We're applying the chain rule here. On the right-hand side, we have 4, which is a constant. And we simply want to differentiate each logarithm. So for the first one we get 1 divided by x + 2 multiplied by x + 2 derivative, which is 1 according to the chain rule, right? For all the logarithms, applying the chain rule basically gives us multiplication by 1 because we have linear functions with a slope of 1. So we get 1 divided by x + 2 plus 1 divided by x minus 3. -1 divided by x minus 1-1 divided by x + 4. And finally, what we can do is solve for the derivative y multiplying both sides by y. Since we know the expression for y, we're going to substitute it immediately. So we're going to get 4 multiplied by y, which is X + 2 multiplied by x minus 3 divided by x minus 1 and divided by x + 4 all raise to the power of 4 multiplied by 1 divided by x + 2 + 1 divided by x minus 3 minus 1 divided by x minus 1 minus 1 divided by x + 4. And this is how we arrive at our final answer. Let's label it and thank you for watching.

In Exercises 25–30, use logarithmic differentiation to find the derivative of y with respect to the appropriate variable.27. y = (((t+1)(t-1))/((t-2)(t+3)))^5, t>2

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In Exercises 25–30, use logarithmic differentiation to find the derivative of y with respect to the appropriate variable.
27. y = (((t+1)(t-1))/((t-2)(t+3)))^5, t>2