The Derivative of a Constant Function: Proving it is Always Zero

If f(x)=k where k is a constant then f'(x)=k for all values of x

To prove that the derivative of a constant function is always zero, let’s start with the definition of a derivative

To prove that the derivative of a constant function is always zero, let’s start with the definition of a derivative.

The derivative of a function f(x) at a point x=a, denoted as f'(a), is defined as the limit of the difference quotient as the interval around a shrinks to zero:

f'(a) = lim(h→0) [(f(a+h) – f(a)) / h]

Now, let’s consider the function f(x) = k, where k is a constant. Substitute f(x) = k into the above definition:

f'(a) = lim(h→0) [(f(a+h) – f(a)) / h]
= lim(h→0) [(k – k) / h]
= lim(h→0) [0 / h]
= lim(h→0) 0
= 0

Since the value of the derivative at any point a is equal to zero, we can conclude that the derivative of a constant function f(x) = k is always zero.

Therefore, f'(x) = k for all values of x.

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