Why is Derivative of 2pie=0 and not 2

albert2008

Dear People,

This might be a really dumb question. Please don't kill me for asking

The derivative of f(x)=2x
F'(x)=2

Can someone elaborate why derivative of 2pie=0
I'm thinking that 2pie is the same as 2x(x replaces pie) and thus the derivative would be 2 but the answer is zero.

Thank you.

CompuChip

Homework Helper
First let me correct a small typo: you meant f'(x) instead of F'(x)... usually, variables and functions in mathematics are case-sensitive, and it is more or less standard notation to use F(x) for a primitive of f(x) (i.e. a function whose derivative F'(x) is f(x)).

To answer your actual question, you should keep in mind that pi is just a number, whose decimal expansion starts with 3.1415..., it is not a variable.
So the derivative of 2x is 2, the derivative of $\sqrt{2}x$ is $\sqrt{2}$, the derivative of 12357832.58312 x is 12357832.58312, and the derivative of $\pi x$ is $\pi$.

In the same way, the derivative of any number is zero, whether that number is 2, 2 pi, square root of (21300 pi^3), or whatever you want.

If you have learned about the geometric meaning of derivative, as slope of a function, this is easy to see. The function f(x) = 2x is a straight line going through the origin and, for example (1, 2). The slope at any point is 2 (just pick an arbitrary point on the line, go one unit to the right, you will have to go two units to the top to get on the line again). If you plot f(x) = 2 pi, that will just be a horizontal line at y = 2 pi for all x. The slope of this line is clearly zero: going one unit to the right you end up at the same height.

2^Oscar

$$\pi$$ is a constant, i.e. a number, so d($$\pi$$)/dx is equal 0.

The difference between $$\pi$$ and x in this context is that x is a variable, so when we differentiate anthing with regard to x we are trying to find the gradient function that will work for any value of x, but because $$\pi$$ is a number (a constant $$\approx$$ 3.14) it has a fixed value and so is not a variable but a constant, like any other number.

Hope this helps to clarify things :)
Oscar

albert2008

Thank you so much for taking time to answer my question. May you always have good health and spirits.

CompuChip

Homework Helper
Thank you so much for taking time to answer my question. May you always have good health and spirits.
No problem. Thank you for your wishes and may you always have more questions to ask :)

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