How do you find the derivative of f(x)= x+ sqrtx ?

2 Answers
Mar 12, 2018

1+(1)/(2sqrt(x))

Explanation:

We note that derivative of sums and differences can be split apart individually. Meaning,

d/dx x + d/dx sqrt(x)

where d/dx x = 1 from the general power rule which states,

d/dx x^n = nx^(n-1)

Continuing on this basis,

sqrt(x) = x^(1/2)

Since we follow the same rules we first bring down the 1/2 and subtract 1 from 1/2 leaving,

d/dx sqrt(x) = 1/2 x^(-1/2)

We further note that we can remove the negative sign of a number by moving it to the opposite of its current location of the form n/d where we now move x^(-1/2) to the denominator resulting in (1)/(2sqrt(x))

Now we simply add the two derivatives together equaling,

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

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

Explanation:

Note the sum rule for derivatives and the power rule:

d/dx (f(x) + g(x)) = f'(x) + g'(x)

As such, d/dx f(x), treating x and sqrt(x) as their own functions in a sense, is...

d/dx (x + sqrt(x))

d/dx x + d/dx sqrt(x)

Using the power rule for the first term...

1 + d/dx x^(1/2)

Using the power rule for the second...

1 + (1/2)x^((1/2)-1)

1 + (1/2)x^(-1/2)

Which can be rewritten as:

1 + 1/(2sqrt(x))