What you are about to learn and what you need to know:
Taking the derivative of a function can be a pain. Using the definition of derivative forces you to write out a rather large equation. This can be very time consuming, and there are quicker ways to do it. One of these ways is a shortcut called the power rule.
This tutorial assumes you understand the basic concept of the derivative. It is also assumed that you have a solid knowledge of algebra.
If you seek a more generic approach to the subject then visit this article, and look at the section on derivatives. This method only works for some of them.
What is the power rule?
The power rule is a way of taking the derivative of an exponential function. For example, 4x^3 is an exponential function that, normally, you would have to plug into the formula for definition of derivative. Someone who knows the power rule can just look at it and know that the derivative is 12x^2.
The power rule states that any term cx^n where “n” is a constant and “c” is the coefficient of “x” has a derivative equivalent to (n*c)x^(n-1). This may sound slightly complicated, but it really is not. It takes the calculus out of taking the derivative and makes it algebra.
Lets use this rule on the previous example, 4x^3. Four is our coefficient of “x” and three is our exponent. So according to the power rule, the derivative would be (3*4)x^(3-1). Lets work this out.
That is precisely the derivative of the function stated previously. If you do a lot of practicing, you too could do derivatives quickly. It is a very cool trick that will help anybody who wants to move up through the calculus track.
How much longer does definition of derivative take?
Definition of derivative takes a whole lot longer to do. If you do not believe the answer the power rule achieved above, then here is the steps to solve it with the definition of derivative.
f ‘(x) = lim h>0 (4(x + h)^3 – 4x^3) / h
f ‘(x) = lim h>0 (4(x^3 + h^3 + 3hx^2 + 3xh^2) – 4x^3) / h (now we expand)
f ‘(x) = lim h>0 (4x^3 + 4h^3 + 12hx^2 + 12xh^2 – 4x^3) / h (distribute the 4)
f ‘(x) = lim h>0 (4h^3 + 12hx^2 + 12xh^2) / h (add like terms)
f ‘(x) = lim h>0 (4h^2 + 12x^2 + 12xh) (divide by h)
f ‘(x) = (4(0)^2+ 12x^2 + 12x(0)) (we can now plug in 0 for h to take the limit)
f ‘(x) = 12x^2 (simplify)
The exact answer the power rule gave you; it just took a whole lot longer.