Are Variables and Constants Distinguishable in Differentiation?

[opticaltempest]

Before we differentiate, we must know whether a variable in that expression represents a constant or variable, correct?

For example, if we have the function,

$$f(x) = r^3 x^2$$

$$f'(x) = \frac{d}{{dx}}\left[ {r^3 x^2 } \right]$$​

(1) Now if $$r$$ represents a variable then,

$$f'(x) = r^3 \cdot \frac{d}{{dx}}\left[ {x^2 } \right] + \frac{d}{{dx}}\left[ {r^3 } \right] \cdot x^2$$

$$f'(x) = 2r^3 x + 3r^2 x^2$$​

But if we know $$r$$ to be a constant then,

$$f'(x) = 2r^3 x$$​

So it seems like it is very important to know exactly what the variable represents in a function. Am I correct? In my current calculus textbook, I don't think this topic was covered thoroughly enough.

EDIT: On line labeled (1) the word "constant" was replaced by the intented word, "variable". My mistake...

 

[benorin]

Now if $$r$$ represents a function of x then,

$$f'(x) = r^3 \cdot \frac{d}{{dx}}\left[ {x^2 } \right] + \frac{d}{{dx}}\left[ {r^3 } \right] \cdot x^2$$

which, by the chain rule, is

$$f'(x) = 2r^3 x + 3r^2 r^{\prime} x^2$$​

But if we know $$r$$ to be a constant then, $$r^{\prime}=0$$

$$f'(x) = 2r^3 x$$​

 

[opticaltempest]

Could you elaborate more on what is meant by "r is a function of x"?

Is there a difference in me saying f(x) vs f(r,x) ?

Since I said f(x) can I think of it like,

"r doesn't allow for any inputs, only x allows inputs. Therefore the value of r will depend upon
what value of x you put in." ?

By me saying f(x)=r*x does it mean that r MUST be some defined constant since it cannot have an input?

 

[HallsofIvy]

If r represents a variable, indpendent of x, then it would make no sense to write f(x)= r³x²- it would have to be f(x,r)= r³x².

If you suspect that r is a function of x (for example that r= cos(x) so that f(x)= r³x²= cos³(x)x² and so f'(x)= -3x²sin²(x)+ 2xcos³(x)), then you can "cover all bets" by writing f'(x)= 3r²x²r'(x)+ 2r³x. In that case, if it happens that r is a constant, r'(x)= 0 giving you f'(x)= 2r³x which is correct.

Generally, whether or not a letter represents a variable or a constant should be clear from the context.

 

[matt grime]

Translation: if something isn't what we supposed it was then its behaviour may change? Answer: Yes. That has nothing to do with maths though.