Using the limit definition, how do you differentiate $f(x)=(x+1)^(1/2)$ ?

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Answer 1 · 2018-05-14T15:53:53

$d/dx ( sqrt(x+1) ) = 1/(2sqrt(x+1))$

Based on the definition of derivative:

$(df)/dx = lim_(h->0) (f(x+h)-f(x))/h$

For $f(x) = (x+1)^(1/2) = sqrt(x+1)$ we have then:

$d/dx ( sqrt(x+1) ) = lim_(h->0) ( sqrt(x+h+1)-sqrt(x+1))/h$

Now multiply numerator and denominator by the quantity:

$sqrt(x+h+1) + sqrt(x+1)$

and use the algebraic identity:

$(a-b)(a+b) = a^2-b^2$ :

$d/dx ( sqrt(x+1) ) = lim_(h->0) (( sqrt(x+h+1)-sqrt(x+1))/h )((sqrt(x+h+1) + sqrt(x+1))/(sqrt(x+h+1) + sqrt(x+1)))$

$d/dx ( sqrt(x+1) ) = lim_(h->0) ( (x+h+1)-(x+1))/(h(sqrt(x+h+1) + sqrt(x+1))$

$d/dx ( sqrt(x+1) ) = lim_(h->0) cancel(h)/(cancel(h)(sqrt(x+h+1) + sqrt(x+1))$

$d/dx ( sqrt(x+1) ) = lim_(h->0) 1/(sqrt(x+h+1) + sqrt(x+1))$

$d/dx ( sqrt(x+1) ) = 1/(2sqrt(x+1))$

Using the limit definition, how do you differentiate f(x)=(x+1)^(1/2)f(x)=(x+1)^(1/2)?