Of all the derivative short-cuts, the most often used is the POWER RULE.
It can take two forms depending on whether we are using functional notation or dependent variable notation.
For functional notation, we want to take the derivative of a function in the form f(x)=[g(x)]n . The resultant derivative would be f'(x)=n [g(x)](n-1) g'(x). In words, “take the power and multiply the expression to one power less (subtract one from the power) then multiply by the derivative of the expression being raised to the power (derivative of the expression inside).
The case for dependent variable notation would look like this: If y=un , u is a function of x, then the derivative, dy/dx=nu(n-1) du/dx. This notation is more concise and easier to remember.
In the cases where g(x) or u are very simple (in the form of x plus or minus a constant), the derivatives g'(x) or du/dx will equal 1. In those cases you don’t have to be concerned about finishing off the rule. These we will call the simple cases of the power rule.
In many cases the g(x) or u could be quite complicated, so, the derivatives at the end would be complicated as well.
Now, examples of the power rule using functional notation.
Example 1: Let f(x)=(7x2 -5x+3)5 , then f'(x)= 5(7x2 -5x+3)4 (14x-5)
Example 2 (simple case): Let f(x)=(x+20)-3 , then f'(x)=-3(x+20)-4 (be careful subtracting 1 from a negative number. (it becomes more negative).
The same examples using dependent variable notation would look like this:
Example 1: Let y=(7x2 -5x+3)5 , then dy/dx= 5(7x2 -5x+3)4 (14x-5)
Example 2: Let y=(x+20)-3 , then dy/dx=-3(x+20)-4
The power rule can be used for all NUMERICAL powers, negatives and fractions included.
Note: Fractional powers are particularly tricky, so be careful here. For example, subtracting 1 from ˝ gives -1/2. Let 1 take a fractional form (any number divided by itself). Subtracting 1 from -5/3 gives -8/3. In the first case, let 1=2/2 & 1=3/3 in the 2nd case.
It can not be used for powers containing a variable such as y=xx or generally, y=uv where u and v are functions of x. In those cases, an advanced technique is used (log differentiation).