Power rule (ax^n)
There isn't a lot of difference between the functions $x^2$x2 and $3x^2$3x2 for example, except that we know the $3$3 has the effect of dilating the graph (in this case making it steeper). This means that the value of the $3$3 must have an affect on the derivative of the graph.
Functions of the type $ax^n$axn are also called power functions, a slight variation of the power rule can be used to find the derivative.
To determine the variation we will use first principles.
| $f(x)$f(x) | $=$= | $ax^n$axn |
| $f(x+h)$f(x+h) | $=$= | $a(x+h)^n$a(x+h)n |
| $f'(x)$f′(x) | $=$= |  |
| $f'(x)$f′(x) | $=$= |  |
From here we need to work through some pretty intense algebra, the full proof is here. You may or may not need to know and follow the proof, if you don't, the key take away is here.
That for functions of the form $f(x)=ax^n$f(x)=axn the derivative is $f'(x)=nax^{n-1}$f′(x)=naxn−1 and this is called the power rule.
Examples
Find the derivatives of
a)$f(x)=3x^2$f(x)=3x2, $f'(x)=6x$f′(x)=6x
b) $g(m)=2m^4$g(m)=2m4, so $g'(m)=8m^3$g′(m)=8m3
c) $h(t)=-3t^{\frac{3}{2}}$h(t)=−3t32, so $h'(t)=\frac{-9}{2}t^{\frac{1}{2}}$h′(t)=−92t12
Be careful with these fractional ones - they are a common source of student errors.
d) $g(x)=\frac{2}{x^4}$g(x)=2x4, firstly we need the function in power form, so we convert it and get that $g(x)=2x^{-4}$g(x)=2x−4. Now we can use the power rule and see that $g'(x)=-8x^{-5}$g′(x)=−8x−5 Remember that $-4-1=-5$−4−1=−5. A common mistake here is to arrive at a power of $-3$−3.
For a function $f(x)=ax^n$f(x)=axn, the derivative $f'(x)=nax^{n-1}$f′(x)=naxn−1
$n$n and $a$a can be positive or negative, integer or fraction
Examples
Question 1
Differentiate $y=5x^5$y=5x5
Question 2
We want to differentiate $y=\frac{7}{x}$y=7x.
First rewrite the function in negative index form.
Now find the derivative, giving your final answer with a positive index.
Question 3
Differentiate $y=\frac{14}{\sqrt{x}}$y=14√x. Express your answer in surd form.