Question

Find the derivative of the function h(x) = √(ln x - 2x - 3).

h'(x) = ________________________________

Answer 1

The derivative h'(x) of the function h(x) = √(ln x - 2x - 3) is found using the chain rule and is (1/x - 2)/(2√(ln x - 2x - 3)).

Step-by-step explanation:

To find the derivative of the function h(x) = √(ln x - 2x - 3), we will use the chain rule and the derivative rules for natural logarithms and polynomial functions. First, let's set u(x) = ln x - 2x - 3. The derivative of the function inside the square root, u'(x), is 1/x - 2 since the derivative of ln x is 1/x and the derivative of 2x is 2. Next, applying the chain rule we get:

h'(x) = ½(u(x))^{-½} · u'(x)

Substituting u(x) and u'(x) into the equation, we have:

h'(x) = ½(ln x - 2x - 3)^{-½} · (1/x - 2),

which simplifies to:

h'(x) = (1/x - 2)/(2√(ln x - 2x - 3))