Derivatives: Why The Derivative Of A Constant Value Is Always Zero

d/dx[c]

0

The derivative of a constant value, such as c, with respect to any variable, including x, is always zero.

This can be demonstrated using the definition of a derivative:

lim h->0 [f(x+h) – f(x)]/h

If f(x) = c, then f(x+h) = c, since c is a constant that does not change with the addition of h. Substituting these values into the definition of the derivative, we get:

lim h->0 [c – c]/h

Simplifying this expression:

lim h->0 0/h

Which evaluates to zero.

Therefore, the derivative of a constant value c with respect to any variable x is zero:

d/dx[c] = 0.

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