LIMIT DEFINITION OF THE DERIVATIVE

We can find the derivative of a function f(x) by evaluating the following limit.

This is called limit definition of the derivative.

Find the derivative of each of the following functions using limit definition of the derivative.

Example 1 :

f(x) = xⁿ

Solution :

Let y = x + h. Then, h = y - x.

When h ---> 0, y ---> x.

From some standard results of limits,

Example 2 :

f(x) = 2x + 3

Solution :

Example 3 :

f(x) = x² + 3x + 1

Solution :

Example 4 :

f(x) = e^x

Solution :

From some standard results of limits,

Example 5 :

f(x) = ln(x)

Solution :

Let y = ʰ⁄ₓ. Then, h = xy.

When h ---> 0, y ---> 0.

From some standard results of limits,

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