How do you find the derivative of #(x^2-2)/(x)#?

Question

Evelyn Agard · Accepted Answer

At first sight we might think of using the quotient rule, but I'd rather operate a division here and simplify things for us. That'll be not only faster but show knowledge on derivation. We see this function #(x^2-2)/x# can be rewritten as #x^2/x-2/x# In turn, it can be rewritten as #x-2x^-1# Now, derivating this sounds easier, doesn't it? #(dy)/(dx)=1-(-2x^-2)#
#(dy)/(dx)=1+2/x^2#

Ezra Adams · Answer

undefined To find the derivative of $\frac{{x^2 - 2}}{{x}}$, you can use the quotient rule. The quotient rule states that if you have a function $f(x)$ divided by another function $g(x)$, then the derivative is given by $\frac{{f'(x)g(x) - f(x)g'(x)}}{{[g(x)]^2}}$. Applying this rule to the given function, where $f(x) = x^2 - 2$ and $g(x) = x$, we get:

$$f'(x) = 2x \text{ (derivative of } x^2 - 2)$$
$$g'(x) = 1 \text{ (derivative of } x)$$

Now, substituting these into the quotient rule formula:

$$\frac{{(2x)(x) - (x^2 - 2)(1)}}{{(x)^2}}$$

Simplify this expression:

$$\frac{{2x^2 - x^2 + 2}}{{x^2}} = \frac{{x^2 + 2}}{{x^2}}$$

This is the derivative of the given function.