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
derivative

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.
graphing derivatives of functions

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.

what are derivatives


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) = 4x2:

The difference f(x + h) – f(x) = 4 [(x + h)2 – x2] = 4 [x2 + 2xh + h2 – x2] = 4 [2xh + h2]. Thus derivative can be found by g'(x) = 4 [(2xh + h2)/h]h->0 = 4(2x + 0) = 8x. Thus the answer is 8x.



More topics in Derivative
Average Velocity The Derivative Function
Derivatives of Power Functions Derivatives of Polynomial Functions
Derivatives of Exponential Functions Instantaneous Rate of Change
Rate of Change Rate of Change Formula
Constant Rate of Change
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