Find the derivatives of the following functions from first principles
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Find the derivative of the following functions from first principle. (i) x^3 – 27 (ii) (x – 1) (x – 2) (iii) 1/x^2 (iv) (x+1) / (x-1)
Answered by admin @
Given : (i) x^3 – 27 (ii) (x – 1) (x – 2) (iii) 1/x^2 (iv) (x+1) / (x-1)
To find : derivative of the functions from first principle
Solution:
f'(x) = Lim h→0 (f(x + h) - f(x)) / h
f(x) = x³ - 27
f'(x) = Lim h→0 ((x + h)³ - x³)/h
=> f'(x) = Lim h→0 (x³ + h³ + 3x²h + 3xh² - x³)/h
=> f'(x) = Lim h→0 ( h³ + 3x²h + 3xh² )/h
=> f'(x) = Lim h→0 h²+ 3x² + 3xh
=> f'(x) = 0 + 3x² + 0
=> f'(x) = 3x²
(ii) f(x) = (x – 1) (x – 2)
f'(x) = Lim h→0 ( (x + h - 1)(x + h - 2) - (x - 1)(x - 2) ) /h
=> f'(x) = Lim h→0 ( -3x - 3h + 2 + x² + h² + 2xh - x² +3x - 2) /h
=> f'(x) = Lim h→0 ( - 3h + h² + 2xh ) /h
=> f'(x) = Lim h→0 ( - 3 + h + 2x )
=> f'(x) = -3 + 0 + 2x
=> f'(x) = 2x - 3
f(x) = 1/x^2
f'(x) = Lim h→0 (1/(x + h)² - 1/x²)/h
=> f'(x) = Lim h→0 ( x² - (x + h)²)/ (x + h)²x²h
=> f'(x) = Lim h→0 ( -h² - 2xh )/ (x + h)²x²h
=> f'(x) = Lim h→0 ( -h - 2x )/ (x + h)²x²
=> f'(x) = ( 0 - 2x )/ (x + 0)²x²
=> f'(x) = - 2/x³
f(x) = (x+1) / (x-1)
f'(x) = Lim h→0 ((x + h + 1)/(x + h - 1) - (x + 1)/(x - 1) )/h
=> f'(x) = Lim h→0 ( x² + xh + x - x - h - 1 - (x² + xh - x + x + h - 1) ) /((x + h - 1) (x - 1) )h
=> f'(x) = Lim h→0 ( -2h ) /((x + h - 1) (x - 1) )h
=> f'(x) = Lim h→0 ( -2 ) /((x + h - 1) (x - 1) )
=> f'(x) = ( -2 ) /((x - 1) (x - 1) )
=> f'(x) = -2/(x - 1)²
Learn more:
Find the derivative of f(x)=sin4x from 1st principle - Brainly.in
Find the derivative by first principle: - Brainly.in
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