DERIVATIVE OF LNX BY FIRST PRINCIPLE

Formula to find the derivative of a function f(x) by first principle.

This is also called limit definition of the derivative.

Let

f(x) = lnx

Derivative of lnx using first principle :

Let y = ʰ⁄ₓ. Then, h = xy.

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

From some standard results of limits,

Solved Problems

Find f'(x) in each case.

Problem 1 :

f(x) = ln(2x + 3)

Solution :

We already know the derivative of lnx, which is ¹⁄ₓ. We can find the derivative of ln(2x + 3) using chain rule.

Problem 2 :

f(x) = ln(x2 + 3x + 2)

Solution :

Problem 3 :

f(x) = ln(ex)

Solution :

Method 1 :

Method 2 :

f(x) = ln(ex)

Using the Power Rule of logarithms,

f(x) = xln(e)

We know that ln(e) is a natural logarithm with base e.

f(x) = xlne(e)

f(x) = x(1)

f(x) = x

f'(x) = 1

Problem 4 :

f(x) = ln(√x)

Solution :

Method 1 :

Method 2 :

Problem 5 :

f(x) = ln(sinx)

Solution :

Problem 6 :

f(x) = ln(cosx)

Solution :

Problem 7 :

f(x) = ln(tanx)

Solution :

Problem 8 :

f(x) = ln(cscx)

Solution :

Problem 9 :

f(x) = ln(secx)

Solution :

Problem 10 :

f(x) = ln(cotx)

Solution :

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