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Find the derivative of y=c from first principles.

May 12, 2026Mathematics

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To find the derivative of $y=c$ from first principles, where $c$ is a constant, we use the definition:

f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}

Identify $f(x)$ and $f(x+h)$ .
Given $f(x) = c$ . Since $c$ is a constant, its value does not change with $x$ .
Therefore, $f(x+h) = c$ .
Substitute $f(x)$ and $f(x+h)$ into the first principles formula.
$f'(x) = \lim_{h \to 0} \frac{c - c}{h}$
Simplify the numerator.
$f'(x) = \lim_{h \to 0} \frac{0}{h}$
Evaluate the limit.
For any $h \neq 0$ , $\frac{0}{h} = 0$ . The limit of 0 as $h$ approaches 0 is 0.
$f'(x) = 0$
The derivative of $y=c$ is $\boxed{0}$ .

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