Differentiation Formulas — Question 2

Question 2

Let $$f(x) = \sqrt{x^2 + 3x} + \frac{1}{\sqrt{x^2 + 3x}}.$$

• Compute $f'(x)$ using differentiation rules.

• Determine all points where the derivative does not exist.

Question 2 - Solution

Domain. Set $u=x^2+3x=x(x+3)$. The reciprocal square root requires $u>0$, so

$$D=(-\infty,-3)\cup(0,\infty).$$

Differentiate.

$$f=u^{1/2}+u^{-1/2},\qquad \boxed{f'(x)=\frac{(2x+3)(x^2+3x-1)}{2(x^2+3x)^{3/2}}},\quad x\in D.$$

The derivative exists at every point of the function’s domain. On $[-3,0]$, the function itself is undefined; these are not nondifferentiable points of its graph.