Find the derivative of the function f(x) = x+ sqrt(X) using the definition of derivative

Question

Please help me finding the derivative of f(x) using the definition of derivative 

Answer 1

Lim h-> 0  Of   f(x+h) - f(x)/h

 

I'm going to simplify the right side first and apply the limit last. Also, some textbooks use Δx instead of h. This makes no difference.

 

Setting up the formula using the function provided gives:

x+h + sqrt(x+h) - (x + sqrt(x)/ h.     note: the entire numerator is over h

 

simplifying the numerator leads to cancellation of x and we have :

 

h + sqrt(x+h)  - sqrt(x) /h   Multiply by the conjugate

 

h + sqrt(x+h) - sqrt(x)/h   * sqrt(x+h) + sqrt(x)/sqrt(x+h) +sqrt(x)

 

=hsqrt(x+h)+hsqrt(x) + x + h+ sqrt(x(x+h) - sqrt(x(x+h) - x/h(sqrt(x+h)+sqrt(x)

 

= hsqrt(x+h)+hsqrt(x)+h/h(sqrt(x+h)+sqrt(x)

 

factor h out numerator and cancel with h in denominator to obtain

 

sqrt(x+h)+sqrt(x)+1/sqrt(x+h)+sqrt(x).   Apply lim h-> 0

 

f'(x) = 1+2sqrt(x)/2sqrt(x) or

 

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

 

 

 

Answer 2

Arianna,

 

The trick here is that the square root of x is actually defined to be a exponent power of x: x^(1/2). You know that to take the derivative, you multiply by the exponent power to find the coefficient and subtract one from the exponent for each term. This means f'(x)=1x^(1-1)+(1/2)x^(-1/2) or 1+(1/2)x^(-1/2).

 

Some people rewrite this to be f'(x)=1+1/(2√x) but this is not necessary. Hope that helps!