In Exercises 115–126, use logarithmic differentiation or the m...

Question

In Exercises 115–126, use logarithmic differentiation or the method in Example 6 to find the derivative of y with respect to the given independent variable.

117. y = (√t)ᵗ

Accepted Answer

Welcome back everyone. Use logarithmic differentiation to find Dy divided by dx for Y equals square root of 2x, raise to the power of x. So for this problem, let's begin by rewriting the function. We have y equals. What we can do is simply rewrite it as 2x, raise to the power of 1/2, right, because we have square root, and then raise to the power of x. And now combining our exponents, this is basically 2 x raised to the power of x divided by 2. And now we're using logarithmic differentiation, which essentially means that we're going to take a line of both sides. So LN of Y is going to be equal to LN of 2 X raise the power of x divided by 2, and we can bring down the exponent using the power rule of logarithms, right? And from here we're going to differentiate both sides. So let's show that we're taking the derivative of the left-hand side and the right hand side. What we are going to do is simply apply the chain rule on the left side. We get 1 divided by y multiplied by y derivative because y is a function of x, and on the right hand side we want to apply the product rule. So that's going to be the derivative of x divided by 2 multiplied by ln of 2 x plus x divided by 2 multiplied by ln of 2 x derivative. So now we are going to get y divided by y equals 1/2 ln of 2 x plus x divided by 2 multiplied by 1 divided by 2 x multiplied by 2 x derivative which is 2, right, using the chain rule. And now let's go ahead and simplify. So now 2 divided by 2 gives us 1. We can eliminate those, and x divided by x, well, this is 1 as well. We can cancel those out. So now y divided by y. is equal to 1/2 ln of 2 x plus 1/2. Now from here, we're going to solve for Y because we want to identify the derivative and Y is going to be equal to 1/2. We can factor out 1/2, right? Multiplied by y and in parentheses we are going to have ln of 2 x plus 1. Now let's remember that y is equal to. Two eggs Raises the power of x divided by 2, so we're going to substitute. So that y is equal to, we can also rewrite one half as 2 raises the power of -1. Now y is. 2 x raised to the power of x divided by 2, we can distribute the exponent and that would be 2 to the power of x divided by 2. Right, Multiplied by x raises the power of x divided by 2, and we are multiplying by our factor ln of 2 x plus 1. Using the properties of exponents, we can rewrite y as 2, raise to the power of x divided by 2 minus 1. Followed by. X raises the power of x divided by 2, which can be rewritten as the square root of x. Raise the power of x. Followed by our factor ln of 2 x plus 1, and this is our final answer for this problem. Thank you for watching.

In Exercises 115–126, use logarithmic differentiation or the method in Example 6 to find the derivative of y with respect to the given independent variable.117. y = (√t)ᵗ

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In Exercises 115–126, use logarithmic differentiation or the method in Example 6 to find the derivative of y with respect to the given independent variable.
117. y = (√t)ᵗ