# How do you use the definition of a derivative to find the derivative of f(x)=-3x?

Nov 22, 2017

The derivative is equal to $- 3$

#### Explanation:

Strictly using the limit definition of the derivative:

The limit definition of the derivative looks like this:
${\lim}_{h \to 0} \frac{f \left(x + h\right) - f \left(x\right)}{h}$

If we replace $f \left(x\right)$ with $- 3 x$, we get:
${\lim}_{h \to 0} \frac{- 3 \left(h + x\right) - 3 x}{h} = {\lim}_{h \to 0} \frac{- 3 h \cancel{- 3 x - \left(- 3 x\right)}}{h}$

Simplifying, we get:
${\lim}_{h \to 0} \frac{- 3 \cancel{h}}{\cancel{h}} = - 3$

So, the derivative of $- 3 x$ must be $- 3$.

Using the meaning of the derivative:

We know that the derivative means the rate of change of the function. Graphically, this means that the derivative is the slope of the graph of that function.

Since $- 3 x$ is a first degree polynomial, we know that it will always have the same slope, and therefor the same derivative. We can also verify this by looking at the graph, noticing that it is a straight line:
We can figure out the slope of this line by remembering the general formula for a linear equation:
$y = m x + b$

In our case, $y = - 3 x$, $b = 0$ and $m = - 3$. In simple terms, the $m$ value represents how much the $y$ value increases for every step in the $x$ direction. This is also exactly what the derivative is, measuring the increase. So we can conclude that the derivative of a linear function is equal to it's $m$ value. In our case, that means that the derivative is equal to $- 3$.