Question

In Exercises \(31-42,\) find \(d y / d x\). $$y=(2 x+5)^{-1 / 2}$$

Short answer

The derivative of the function \(y=(2 x+5)^{-1 / 2}\) with respect to 'x' is \(d y / d x = -1(2 x + 5)^{-3 / 2}\).

Step-by-step solution

Identify the Outer and Inner Function

The function to be differentiated can be considered as a composition of two functions. Here, the outer function is \(f(u) = u^{-1/2}\) and the inner function is \(u = g(x) = 2x + 5\).

Differentiate the Outer Function

Differentiate the outer function \(f(u) = u^{-1/2}\) with respect to 'u', giving us \(f'(u) = -1/2 * u^{-3/2}\).

Differentiate the Inner Function

Now differentiate the inner function \(g(x) = 2x + 5\) with respect to 'x', giving us \(g'(x) = 2\).

Apply Chain Rule

According to the chain rule, \(f'(g(x)) = f'(u) * g'(x)\). Substituting the derivatives of the outer and inner functions we get \(f'(g(x)) = -1/2 * (2x + 5)^{-3/2} * 2\).