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,
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(x^{2} + 3x + 2)
Solution :
Problem 3 :
f(x) = ln(e^{x})
Solution :
Method 1 :
Method 2 :
f(x) = ln(e^{x})
Using the Power Rule of logarithms,
f(x) = xln(e)
We know that ln(e) is a natural logarithm with base e.
f(x) = xln_{e}(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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