So, lets see how we can apply the first principle to differentiate functions.
We’ll start with the basic functions, lets take the square function first.

So the derivative of the square function is 2x which is also the slope of the tangent at any point on the curve of the function. So, for a point, x=a, the slope of the tangent would be 2a(putting x=a in 2x) And, since we know the slope of this tangent at a, we can also find its equation. Hence, we can use derivatives to find equations of tangents which is an important Application Of The Derivative.

Now, lets differentiate another function using this principle. We’ll go for the rectangular hyperbola this time.

Now, lets find out a generalization for such functions(polynomial/rational & irrational).

We considered the value (phi) to collectively consist of all the higher powers of the change in x as the terms with higher powers would eventually cancel out while solving the limit.
Hence, this is the actual method for finding derivatives of functions – the first principle. However, in practice, we use properties of derivatives and the basic derivatives of the most common functions to find the derivatives of bigger and more complex functions. First, we must be aware of the derivatives of the major and most common functions.

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