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differentiation by first principle method

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Solution

The explanation is as follows.

We can also define derivative as the rate of change of a given function. It is little different from the average rate of change. Derivative is expressed in terms of the limit of rate of change in some function.The limit of the length of interval approaches to zero. The differentiation may be explained as the optimum linear approximation of a function near given input value.

For Example: The velocity of an object is the derivative of position of an object with respect to time. Also, the derivative of velocity of an object with respect to time is known as acceleration.

There are various formulas for finding derivatives of different types of functions. There is a technique of differentiating almost all the functions. It is called differentiation by first principles.
In order to perform the differentiation of a function by first principles, one requires to follow the steps written below:

Step 1: Suppose the given function be f(x); where x be the independent variable which can be denoted by any other variable as given in the function.

Step 2: Determine the value of f(x + h). This can be done by substituting x + h in place of x in the function f(x).

Step 3: Substitute the value of f(x) and f(x + h) in the formula:

f′(x)=f′(x)=\lim_{h \to 0}$ f(x+h)−f(x)hf(x+h)−f(x)h

Step 4: Solve into the simpler form.

Step 5: Plug the limit of h tends to zero; i.e. put zero in place of h and solve. This given the required derivative.

You can also go through the below link.
Byju's article on derivatives

Hope it answers your question.
All the best!

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