Differentiate the function. Y= √x(x-8) ?

Question

Differentiate the function.  Y= √x(x-8) ?

Isabella Ackerman · Accepted Answer

The answer is: #(x-4)/sqrt(x^2-8x)#. #y=sqrt(x(x-8))rArry=sqrt(x^2-8x)rArr# #y'=1/(2sqrt(x^2-8x))*(2x-8)=(2(x-4))/(2sqrt(x^2-8x))=(x-4)/sqrt(x^2-8x)#.

Emilia Aguilar · Answer

undefined To differentiate the function y = √x(x-8), we can use the product rule and the chain rule.

First, let's rewrite the function as y = x^(1/2) * (x-8)^(1/2).

Using the product rule, the derivative of y with respect to x is given by:

dy/dx = (x-8)^(1/2) * d/dx(x^(1/2)) + x^(1/2) * d/dx((x-8)^(1/2)).

Now, let's find the derivatives of each term using the chain rule:

d/dx(x^(1/2)) = (1/2) * x^(-1/2).

d/dx((x-8)^(1/2)) = (1/2) * (x-8)^(-1/2) * d/dx(x-8).

Since d/dx(x-8) = 1, the derivative simplifies to:

d/dx((x-8)^(1/2)) = (1/2) * (x-8)^(-1/2).

Substituting these derivatives back into the product rule equation, we have:

dy/dx = (x-8)^(1/2) * (1/2) * x^(-1/2) + x^(1/2) * (1/2) * (x-8)^(-1/2).

Simplifying further, we get:

dy/dx = (x-8)^(1/2) / (2√x) + x^(1/2) / (2√(x-8)).

Therefore, the derivative of y = √x(x-8) is:

dy/dx = (x-8)^(1/2) / (2√x) + x^(1/2) / (2√(x-8)).

Christopher Allers · Answer

undefined To differentiate the function $ y = \sqrt{x(x-8)} $, use the chain rule and the power rule:

1. First, rewrite the function as $ y = (x(x-8))^{1/2} $.
2. Apply the chain rule: $ y' = \frac{1}{2} (x(x-8))^{-1/2} \cdot (2x - 8) $.
3. Simplify to get the derivative: $ y' = \frac{x - 4}{\sqrt{x(x-8)}} $.