# How do you find the derivative of f(x)=x^2-5x+3?

Apr 9, 2018

$f ' \left(x\right) = 2 x - 5$

#### Explanation:

The derivative of a function is the instantaneous rate of change of the function at a given point. But we have a function, so we want to find the rate of change throughout. We have two ways of doing this, and I'll show you both ways.

Method 1: The Power Rule

This method only works for polynomials (${x}^{n}$). We have a saying that helps remember the power rule: "Bring the power forward, take one off the power." Essentially, that means the following:

$\frac{d}{\mathrm{dx}} \left({x}^{n}\right) = n {x}^{n - 1}$

So we can apply this to your function. But we also have a constant. Remember that adding or subtracting a constant yields only a vertical shift, not a stretch. Therefore, it has no effect on the rate of change, so we can also say:

$\frac{d}{\mathrm{dx}} \left(n\right) = 0$

We can use these two rules together to get the answer:

$\frac{d}{\mathrm{dx}} \left({x}^{2} - 5 x + 3\right) = 2 x - 5$

Method 2: First Principles

First Principles is a method that allows you to take the derivative of any function. It comes from the slope formula ($\frac{r i s e}{r u n}$). We want to get to an infinitely small interval on the graph, so we can use the slope formula to come up with the slope at a given point of the function:

${\lim}_{h \to 0} \left(\frac{f \left(x + h\right) - f \left(x\right)}{h}\right)$

So let's plug in your function:

${\lim}_{h \to 0} \left(\frac{\left({\left(x + h\right)}^{2} - 5 \left(x + h\right) + 3\right) - \left({x}^{2} - 5 x + 3\right)}{h}\right)$

Since we can't divide by 0, we need to get the h out of the denominator. So let's expand the binomial.

${\lim}_{h \to 0} \left(\frac{\left({x}^{2} + 2 x h + 2 {h}^{2} - 5 x - 5 h + 3\right) - \left({x}^{2} - 5 x + 3\right)}{h}\right)$

And let's combine all the like terms (cancelling out most of the x values):

${\lim}_{h \to 0} \frac{\left(2 x h + 2 {h}^{2} - 5 h\right)}{h}$

Now, every term as an h somewhere in it. So we can divide by h:

${\lim}_{h \to 0} \left(2 x + 2 h - 5\right)$

And now as the limit approaches 0, 2h reaches 0, and therefore, we are left with the final answer:

$f ' \left(x\right) = 2 x - 5$