The Power Rule

Let’s compute some basic derivatives from the definition:

$$f(x)=x$$ (We can guess the answer here from the slope-definition, but its good to confirm!)

$$f(x)=x^2$$ (Here we could guess the shape from drawing the graph and sketching the derivative, but we didn’t know the exact equation of the curve without doing the calculation)

$$f(x)=x^3$$

All of the three above examples involve expanding out the numberator and then evaluating the limit.
But we can look at other functions where the cacluation is also rather simple, but proceeds by different means: $$f(x)=1/x$$ Here you need to combine the fractions in the numerator before computing the limit.

$$f(x)=\sqrt{x}$$ Here you need to ‘rationalize the numerator’ to be able to make progress!

A pattern among these cases:

If we write each of the above functions as $x$ to some power, a pattern emerges: $$(x{)}^{\prime }=1$$ $$(x^2{)}^{\prime }=2x$$ $$(x^3{)}^{\prime }=3x^2$$ $$(x^{-1}{)}^{\prime }=-x^{-2}$$ $$(x^{1/2}{)}^{\prime }=\frac{1}{2}{x}^{-\frac{1}{2}}$$

In each case, differentation managed to “bring the original power down to multiply as a coefficient”, and then decrease the power by 1. In fact, this rule holds for all real number exponents, and is called the power rule:

$$(x^n{)}^{\prime }=n{x}^{n-1}$$