Question

Use the limit definition to find the derivative of the function. $$ f(x)=-2 $$

Short answer

The derivative of the function \(f(x)=-2\) is zero.

Step-by-step solution

Substitute the function into the limit definition

We start by substituting \(f(x)=-2\) into the limit definition of the derivative. This gives us\[f'(x) = \lim_{h \rightarrow 0} \frac{ f(x+h) - f(x) }{ h } = \lim_{h \rightarrow 0} \frac{ -2 - (-2) }{ h } = \lim_{h \rightarrow 0} \frac{ 0 }{ h }\]

Calculate the Limit

Next we calculate the limit. Since the numerator is zero, the whole limit is zero. It doesn’t matter what value \(h\) approaches. So we have\[f'(x) = \lim_{h \rightarrow 0} \frac{ 0 }{ h } = 0\]