# Lets study formal definition of a derivative

We know that the value of limit of a function also changes with change in input. Hence **derivatives** can be expressed in terms of limits which becomes **formal definition of a derivative**. This can be expressed as

It also helps to find the slope at any given point on the curve.

Thus **derivatives calculus **is derived from the limits concepts. Many more **derivative** formulas can be found out using this concept. Now that we have seen **what is derivative **and **derivatives definition**, we will be **graphing derivatives**.

**Learn graphing derivatives of functions:**

** **Consider the function g(x) graphed below. We will be graphing its derivative and list out the values in **derivative table**.

We know that the derivative g'(x) can be found out by taking the slope of tangent at (x, g(x)) according to **derivative definition calculus**. Now we will mention values of g'(x) corresponding to x and plot a graph for same. Note that this one will be a rough graph as its really hard measuring the slope without the ruler scales and grid lines. But it will give you an idea of what is actually taking place.

x |
0 |
0.5 |
1 |
1.5 |
2 |
2.5 |

g'(x) |
3 |
0 |
-4 |
-3 |
0 |
1 |

Based on the above table, the graph for g'(x) is given below that explains **what are derivatives **in better sense. Note that corresponding points are plotted as given in the table.

Now that we have learned **derivative definition **and graphed the derivatives, lets have a couple of **derivatives examples**.

(1) Find the derivative of 3x:

Lets do it using limits definition. We have f(x + h) – f(x) = 3x + 3h – 3x = 3h. Hence derivative is given by [3h/h]_{h->0} = 3. Thus the answer is 3.

(2) Evaluate g'(x) if g(x) = 4x^{2}:

The difference f(x + h) – f(x) = 4 [(x + h)^{2} – x^{2}] = 4 [x^{2} + 2xh + h^{2} – x^{2}] = 4 [2xh + h^{2}]. Thus derivative can be found by g'(x) = 4 [(2xh + h^{2})/h]_{h->0} = 4(2x + 0) = 8x. Thus the answer is 8x.

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