Home > Calculus > First Principles Example 3: square root of x

How do you solve this problem step by step?

Find the second derivative #f(x)=x^2+1/x#

Answer 1

Samantha Adkison

#f'(x) = 2x-1/x^2#
#f''(x) = 2(1+1/x^3)#

#f(x) = x^2+1/x#

First remember that #1/x =x^-1#

#:. f(x) = x^2+x^-1#

#f'(x) = d/dx(x^2) + d/dx (x^-1)#

Now we can apply the power rule to both terms.

The power rule states that #d/dx x^n = nx^(n-1)#

Hence, #f'(x) = 2x^1 + (-1)x^-2 -> 2x - 1/x^2#

Similarly, #f''(x) = 2x^0 - (-2)x^-3 -> 2+2/x^3 = 2(1+1/x^3)#

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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