# Definition of derivative

## What is a Derivative?

The derivative is basically the slope. So the derivative of a function is the slope of the function for a given point. We also denote the derivative as $\frac{dy}{dx}$. Of course, some functions cannot be differentiated for a specific point. Take the example of the derivative of $\frac{1}{x}$. When $x=0$, we know that the function is undefined. Hence, we also know that the derivative is also undefined.

## How to find the Derivative?

There are two ways to find the derivative:

1. Use the definition of derivative

2. Use the derivative chart and the derivative rules

We will explain both ways.

## Definition of Derivative

Recall that we can find the derivative by using the definition of derivative shown below:

• Let’s look at some examples of using the definition of derivative.

• First, find the derivative of a constant. Let $f(x) = c$, where $c$ is a constant. Using the definition of derivative will give us:

Hence, the derivative of a constant will always be $0$.

• What about the derivative of square root? Let $f(x) = \sqrt{x}$. Then using the definition of derivative will give us:

• This one is a little tricky because we need to manipulate the equation a little bit. What were going to do is something called a conjugate. We’re going to multiply both the numerator and denominator by $\sqrt{(x+h)} + \sqrt{x}$. In other words,

• Simplifying will give us:

• Notice that the $h$’s in the numerator and denominator can be cancelled out, leaving us

• Now we can finally take the limit and plug in $h = 0$, which gives us:

Hence, the equation above is the derivative of the square root of $x$.

Now let’s try something similar and take the derivative of an absolute value.Let $f'(x) = |x|$. What we need to notice here is that $|x|$ can also be written as $\sqrt{x^{2}}$ . Hence, using the definition of derivative gives us

• Again, we are going to do the conjugate. So

• Simplifying the numerator will give us:

• Cancelling the $h$’s gives

• Now taking the limit (set $h=0$), and simplifying will give us:

• Multiplying the numerator and denominator by $\sqrt{x^{2}}$ will give us:

• Since we know from the beginning that $|x| = \sqrt{x^{2}}$, then we finally say that

• Now why don’t we try something harder and take the derivative of an exponential function? Let $f(x) = e^{x}$. Then using the definition of derivative gives us:

• Notice that $e^{(x+h)} = e^{x}e^{h}$. Hence we can rewrite our equation to

• Now we can factor out $e^{x}$ and pull it out of the limit, giving us:

• Now the hard part here is taking this limit. In order to continue on from here, we need to look at the definition of e. Notice that

• Taking both sides to the power of $h$ will give us:

• We are going to substitute this $e^{h}$ into the original equation we had earlier, which gives us:

• Doing a little bit of algebra will give us that

• and so we can conclude that

If you want to try harder problems, take the derivative of the natural log. In other words, find the derivative of $\ln x$! If you want the solution, then look at the link below.

http://math2.org/math/derivatives/more/ln.htm

## Derivative Chart

Here is a table of common derivatives shown below

These can be used when applying the derivative rules.

## Derivative Rules

Now derivative rules are very useful when we’re trying to take the derivative of uncommon functions. For example, we know that the derivative of $e^{x}$ is $e^{x}$, but what about the derivative of $e^{-x}$? What about the derivative of $e^{2x}$? This is where we introduce the chain rule. The chain rule says the following:

• Let’s use this to take the derivative of $e^{-x}$? Let $g(x) = -x$, and $f(x) = e^{x}$. Then we can say $f'(x) = e^{x}$, and so $f'[g(x)] = e^{-x}$. Applying the chain rule will give us:

Hence, the derivative of $e^{-x}$ is -$e^{-x}$.

• Now let us take the derivative of $e^{2x}$. If $g(x) = 2x$, and $f(x) = e^{x}$. Then we can say f’(x) = e^{x}, and so $f'[g(x)] = e^{2x}$ Using the chain rule will give us:

• Now there is another derivative rule which lets us take the derivative of a fraction. We call this the quotient rule. The quotient rule says the following:

• For example, let $h(x) = \frac{2^{x}}{x^{3}}$. We set $f(x) = 2^{x}$ and $g(x) = x^{3}$. Know from our derivative chart that $f'(x) = 2^{x} \ln 2$ and $g'(x) = 3x^{2}$. Hence, using the quotient rule will give us the following:

There are also other derivative rules such as the product rule and power rule. So if you want to learn about those as well, we recommend you to click those links.

### Definition of derivative

We have studied the notion of average rate of change thus far, for example, change in position over time (velocity), average change in velocity over time (acceleration) etc. However, what if we are interested in finding the instantaneous rate of change of something? To answer this, we will first learn about the concept of the definition of derivative in this section, as well as how to apply it.