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Question: Consider the following statement If f and g be two functions such that f(x) ≠ 0, g(x) ≠f '(x) ≠0 an...

Consider the following statement

If f and g be two functions such that f(x) ≠ 0, g(x) ≠f '(x) ≠0 and g'(x) ≠0 for all x, then –

(1) (fg)′=f′g′\left( \frac{f}{g} \right)^{'} = \frac{f'}{g'} (2) (fg)′fg=f′f+g′g\frac{(fg)^{'}}{fg} = \frac{f'}{f} + \frac{g'}{g}

(3) (f+g)′f+g=f′f+g′g\frac{(f + g)^{'}}{f + g} = \frac{f'}{f} + \frac{g'}{g} (4) (f/g)′f/g=f′f−g′g\frac{(f/g)^{'}}{f/g} = \frac{f'}{f} - \frac{g'}{g}

Which of these statements are correct?

A

(1) and (2)

B

(2) and (3)

C

(2) and (4)

D

(3) and (4)

Answer

(2) and (4)

Explanation

Solution

(fg)′fg=fg′+gf′fg=g′g+f′f\frac{(fg)'}{fg} = \frac{fg' + gf'}{fg} = \frac{g'}{g} + \frac{f'}{f}

(f/g)′f/g\frac{(f/g)'}{f/g}=gf′−fg′g2f/g\frac{\frac{gf' - fg'}{g^{2}}}{f/g}= gf′−fg′fg\frac{gf' - fg'}{fg}= f′f\frac{f'}{f}– g′g\frac{g'}{g}