How to differentiate an expression with a variable and parameters

Question

I came across the expression of $\sqrt{x^2+a^2}$ with $a$ as a non zero real number parameter. I thought that I would simply treat $a^2$ as simply one integer when doing derivations but my instructor said that I was wrong. Can anyone please help explain why this is the case and inform me how to treat parameters in f(x) expressions? thanks a lot!

Answer 1

The phrase "a non zero real number parameter" is ill-posed. You can either have $a$ as a real number, hence $a\in\mathbb{R}$, which is also called a real constant, or you can have $a = a(x)$ (or also $a = a(x, t, \ldots)$) a real valued non constant function.

If, say, you have to take the derivative of that function you then have different cases.

If $a$ is a real number, that is a constant function (you can always think of $a$ as a function of many variables, but independent from them, even if it's not really useful, like $a(x, y) = \pi$ but it's a possibility), then the derivative won't see that number, namely

$$\dfrac{\text{d}}{\text{d}x} \sqrt{x^2 + a^2} = \dfrac{\text{d}}{\text{d}x}\sqrt{f(x)} \qquad \text{with} \qquad f(x) = x^2 + a^2$$

And when operating the derivative you obtain (following the rule of the derivative of the square root of a function):

$$\dfrac{\text{d}}{\text{d}x} = \dfrac{1}{2\sqrt{x^2 + a^2}}\cdot 2x = \dfrac{x}{\sqrt{x^2 + a^2}}$$

(Remark: $\dfrac{\text{d}}{\text{d}x} \sqrt{f(x)} = \dfrac{1}{2\sqrt{f(x)}}\cdot \dfrac{\text{d}f(x)}{\text{d}x}$).

In this case

$$\dfrac{\text{d}}{\text{d}x} f(x) = \dfrac{\text{d}}{\text{d}x} (x^2 + a^2 )= \dfrac{\text{d}x^2}{\text{d}x} + \dfrac{\text{d}a}{\text{d}x} = 2x + 0$$

since $a$ is indeed a constant function (or, again, a real number).

If instead $a = a(x)$ then when computing the derivative of $f(x)$ (now $f(x) = x^2 + a(x)$, you have to take into account that $a(x)$ is a function (non constant) depending on $x$, hence it will be seen by the derivative operator, whence:

$$\dfrac{\text{d}}{\text{d}x} f(x) = \dfrac{\text{d}}{\text{d}x}(x^2 + a(x)) = \dfrac{\text{d}x^2}{\text{d}x} + \dfrac{\text{d}a(x)}{\text{d}x} = 2x + a'(x)$$

Where $a'(x)$ is a notation standing for $\dfrac{\text{d}a(x)}{\text{d}x}$, which we must leave like that since we have no information on the explicit expression of $a(x)$.

In any case, this is just a long answer which might now be needed, since you did not specify the real nature of your problem (and most of all of your professor's request).