Find the derivative using first principles? : #x+sqrt(x)#

1 Answer
Steve M
Jan 29, 2018

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

Explanation:

Let us define:

# f(x) = x+sqrt(x)#

Then using the limit definition of the derivative, we have:

# f'(x) = lim_(h rarr 0) (f(x+h)-f(x))/h #

# \ \ \ \ \ \ \ \ \ = lim_(h rarr 0) ((x+h)+sqrt(x+h)-x+sqrt(x))/h #

# \ \ \ \ \ \ \ \ \ = lim_(h rarr 0) (h+sqrt(x+h)-sqrt(x))/h #

# \ \ \ \ \ \ \ \ \ = lim_(h rarr 0) 1+(sqrt(x+h)-sqrt(x))/h * (sqrt(x+h)+sqrt(x))/(sqrt(x+h)+sqrt(x))#

# \ \ \ \ \ \ \ \ \ = lim_(h rarr 0) 1+((x+h)-(x))/(h(sqrt(x+h)+sqrt(x))#

# \ \ \ \ \ \ \ \ \ = lim_(h rarr 0) 1+(h)/(h(sqrt(x+h)+sqrt(x))#

# \ \ \ \ \ \ \ \ \ = lim_(h rarr 0) 1+(1)/(sqrt(x+h)+sqrt(x)#

# \ \ \ \ \ \ \ \ \ = 1+(1)/(sqrt(x+0)+sqrt(x)#

# \ \ \ \ \ \ \ \ \ = 1+(1)/(2sqrt(x)#

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