Definition of derivative

Definition of Derivative and Common Derivatives

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 dydx\frac{dy}{dx}. Of course, some functions cannot be differentiated for a specific point. Take the example of the derivative of 1x\frac{1}{x}. When x=0x=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:

  • Formula 1: Definition of Derivative
    Formula 1: Definition of Derivative
  • Let’s look at some examples of using the definition of derivative.

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

    Equation 1: Derivative of constant
    Equation 1: Derivative of constant

    Hence, the derivative of a constant will always be 00.

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

    Equation 2: Derivative of square root pt.1
    Equation 2: Derivative of square root pt.1
  • 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 (x+h)+x\sqrt{(x+h)} + \sqrt{x}. In other words,

    Equation 3: Derivative of square root pt.2
    Equation 3: Derivative of square root pt.2
  • Simplifying will give us:

    Equation 4: Derivative of square root pt.3
    Equation 4: Derivative of square root pt.3
  • Notice that the hh’s in the numerator and denominator can be cancelled out, leaving us

    Equation 5: Derivative of square root pt.4
    Equation 5: Derivative of square root pt.4
  • Now we can finally take the limit and plug in h=0h = 0, which gives us:

    Equation 6: Derivative of square root pt.5
    Equation 6: Derivative of square root pt.5

    Hence, the equation above is the derivative of the square root of xx.

Now let’s try something similar and take the derivative of an absolute value.Let f(x)=xf'(x) = |x|. What we need to notice here is that x|x| can also be written as x2\sqrt{x^{2}} . Hence, using the definition of derivative gives us

  • Equation 7: Derivative of absolute value pt.1
    Equation 7: Derivative of absolute value pt.1
  • Again, we are going to do the conjugate. So

    Equation 8: Derivative of absolute value pt.2
    Equation 8: Derivative of absolute value pt.2
  • Simplifying the numerator will give us:

    Equation 9: Derivative of absolute value pt.3
    Equation 9: Derivative of absolute value pt.3
  • Cancelling the hh’s gives

    Equation 10: Derivative of absolute value pt.4
    Equation 10: Derivative of absolute value pt.4
  • Now taking the limit (set h=0h=0), and simplifying will give us:

    Equation 11: Derivative of absolute value pt.5
    Equation 11: Derivative of absolute value pt.5
  • Multiplying the numerator and denominator by x2\sqrt{x^{2}} will give us:

    Equation 12: Derivative of absolute value pt.6
    Equation 12: Derivative of absolute value pt.6
  • Since we know from the beginning that x=x2|x| = \sqrt{x^{2}}, then we finally say that

    Equation 13: Derivative of absolute value pt.7
    Equation 13: Derivative of absolute value pt.7
  • Now why don’t we try something harder and take the derivative of an exponential function? Let f(x)=exf(x) = e^{x}. Then using the definition of derivative gives us:

    Equation 14: Derivative of e^x pt.1
    Equation 14: Derivative of e^x pt.1
  • Notice that e(x+h)=exehe^{(x+h)} = e^{x}e^{h}. Hence we can rewrite our equation to

    Equation 15: Derivative of e^x pt.2
    Equation 15: Derivative of e^x pt.2
  • Now we can factor out exe^{x} and pull it out of the limit, giving us:

    Equation 16: Derivative of e^x pt.3
    Equation 16: Derivative of e^x pt.3
  • 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

    Equation 17: Derivative of e^x pt.4
    Equation 17: Derivative of e^x pt.4
  • Taking both sides to the power of hh will give us:

    Equation 18: Derivative of e^x pt.5
    Equation 18: Derivative of e^x pt.5
  • We are going to substitute this ehe^{h} into the original equation we had earlier, which gives us:

    Equation 19: Derivative of e^x pt.6
    Equation 19: Derivative of e^x pt.6
  • Doing a little bit of algebra will give us that

    Equation 20: Derivative of e^x pt.7
    Equation 20: Derivative of e^x pt.7
  • and so we can conclude that

    Equation 21: Derivative of e^x pt.8
    Equation 21: Derivative of e^x pt.8

If you want to try harder problems, take the derivative of the natural log. In other words, find the derivative of lnx\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

Derivative Chart
Derivative Chart

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 exe^{x} is exe^{x}, but what about the derivative of exe^{-x}? What about the derivative of e2xe^{2x}? This is where we introduce the chain rule. The chain rule says the following:

  • Definition 1: Chain Rule
    Definition 1: Chain Rule
  • Let’s use this to take the derivative of exe^{-x}? Let g(x)=xg(x) = -x, and f(x)=exf(x) = e^{x}. Then we can say f(x)=exf'(x) = e^{x}, and so f[g(x)]=exf'[g(x)] = e^{-x}. Applying the chain rule will give us:

    Equation 22: Derivative of e^-x
    Equation 22: Derivative of e^-x

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

  • Now let us take the derivative of e2xe^{2x}. If g(x)=2xg(x) = 2x, and f(x)=exf(x) = e^{x}. Then we can say f’(x) = e^{x}, and so f[g(x)]=e2xf'[g(x)] = e^{2x} Using the chain rule will give us:

    Equation 23: Derivative of e^2x
    Equation 23: Derivative of e^2x
  • 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:

    Definition 2: Quotient Rule
    Definition 2: Quotient Rule
  • For example, let h(x)=2xx3h(x) = \frac{2^{x}}{x^{3}}. We set f(x)=2xf(x) = 2^{x} and g(x)=x3g(x) = x^{3}. Know from our derivative chart that f(x)=2xln2f'(x) = 2^{x} \ln 2 and g(x)=3x2g'(x) = 3x^{2}. Hence, using the quotient rule will give us the following:

    Equation 24: Derivative of a fraction
    Equation 24: Derivative of a fraction

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.

Lessons

Notes:
Definition of Derivative
f(x)=h0limf(x+h)f(x)hf'\left( x \right) = \;_{h \to 0}^{\;lim}\frac{{f\left( {x + h} \right) - f\left( x \right)}}{h}
  • 2.
    Definition of derivative with irregular functions
    Find the derivative of the following functions using the definition of derivative.
  • 3.
    Applications to definition of derivative
    Let f(x)=4x13f(x)=4x^{\frac{1}{3}}
Teacher pug

Definition of derivative

Don't just watch, practice makes perfect.

We have over 170 practice questions in Calculus 1 for you to master.