Several ways for finding the derivative of a function

As we know, derivatives are all about change. It is the rate of change of a quantity as compared to the another one according to the derivative of a function definition. Just like velocity is the rate of change of distance with respect to the change in time. Let us consider that y is a function of x, i.e. y = f(x) then dy/dx is called the derivative of a function. According to the definition of the derivative of a function, it gives the rate of change of y with respect to x.

Calculus limits provide basis for finding derivatives of a function. The derivative of function y = f(x) at x = a can be given as if the limits exist. It can be denoted by f' (a). Now consider f(x) for a closed interval [a,b]. Then the rate of change can be given by For the linear functions, this concept holds true for linear functions. But for non-linear functions, it just provides the average rate. Consider a infinitely small point h that is close to value at f(x). Now finding the derivative of a function, it is given by the formula It's easy graphing the derivative of a function

Now we will be graphing the derivative of a function for y = f(x) = ln(x) which comes 1/x as shown below.  Some important derivative function examples

Some of the important derivative function examples are given below to get a quick overview of derivatives.

(1) Find the derivative of x2:

Here f(x) = x2. Hence f(x + h) = (x + h)2 = x2 + 2xh + h2. Taking the difference, f(x + h) – f(x) = h2 + 2xh. Thus f'(x) = [(h2 + 2xh) / h]h->0 = [h + 2x]h->0 = 2x. Thus derivative of x2 comes to be 2x.

(2) Find the f'(x) if f(x) = 3x:

Here f(x) = 3x. Hence f(x + h) = 3x + 3h which yields f(x + h) - f(x) = 3x + 3h – 3x = 3h. Now taking the ratio, f'(x) = [(f(x + h) – f(x)) / h]h->0 = [3h / h]h->0 = 3. Thus derivative of 3x is a constant value 3.

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