What is a symbolic differentiation of simple arithmetic expressions?

Question

I am making some exercises in my programming book and one is the following:

Perform symbolic differentations of simple arithmetic expressions with respect to a single variable.

Can somebody explain this to me and perhaps give me some examples?

Answer 1

Seems to me that the question may have in mind functions involving the 4 basic arithmetic operations, i.e. addition, subtraction, multiplication and division, for 2 different functions of some variable, x.

For example, if we have :

$$ y = f(u, v) $$ where $u \equiv u(x)$ and $v \equiv v(x)$ .

1. $$y = u(x) + v(x)$$ $$ \frac{dy}{dx} = u'(x) + v'(x) $$

2. $$y = u(x) - v(x)$$ $$ \frac{dy}{dx} = u'(x) - v'(x) $$

3. $$y = u(x) * v(x)$$ $$ \frac{dy}{dx} = u(x) * v'(x) + v(x) * u'(x)$$

4. $$y = u(x) / v(x)$$ $$ \frac{dy}{dx} = \frac{v(x) * u'(x) - u(x) * v'(x)}{v(x)^2}$$

But I'm not sure what you mean by "in my programming book" . . .

Are you doing these as a C++/Java programming exercise ? Are you writing method code for evaluating $ \frac{dy}{dx} $ for simple expression functions so that you can then more easily obtain $ \frac{dy}{dx} $ for more complex functions if they can be resolved into arithmetic expressions of simpler ones ?