Question

How do you use the definition of a derivative to find the derivative of `sqrt(2x)`?

Answer 1

`f'(x)=1 / ( sqrt(2x) )`

Explanation:

By definition of the derivative `f'(x)=lim_(h rarr 0) ( f(x+h)-f(x) ) / h`
So with `f(x) = sqrt(2x)` we have;

`f'(x)=lim_(h rarr 0) ( sqrt(2(x+h)) - sqrt(2x) ) / h`
`:. f'(x)=lim_(h rarr 0) ( sqrt(2x+2h) - sqrt(2x) ) / h`
`:. f'(x)=lim_(h rarr 0) ( sqrt(2x+2h) - sqrt(2x) ) / h * ( sqrt(2x+2h) + sqrt(2x) )/( sqrt(2x+2h) + sqrt(2x) )`
`:. f'(x)=lim_(h rarr 0) ( 2x+2h - 2x ) / (h( sqrt(2x+2h) + sqrt(2x) ))`
`:. f'(x)=lim_(h rarr 0) ( 2h ) / (h( sqrt(2x+2h) + sqrt(2x) ))`
`:. f'(x)=lim_(h rarr 0) 2 / ( sqrt(2x+2h) + sqrt(2x) )`
`:. f'(x)=2 / ( 2sqrt(2x) )`
`:. f'(x)=1 / ( sqrt(2x) )`