2.1, The Derivative As a Function

Given a function f(x), the slope of the curve y = f(x) at the point (a, f(a)) is the derivative at a. Given f(x), we can work out formulas for this slope (derivative) and so create a new function,

f '(x)

where the slope of the curve y = f(x) at the point (a, f(a)) is f '(a).

Some basic formulas

We have already worked out a few derivative formulas.

1. The derivative of  x^n is nx^(n-1) (the power rule).
2. The derivative of sin x is cos x; the derivative of cos x is –sin x.
3. The derivative of e^x is e^x.

Here, from my lecture notes, is a summary:

We will combine these basic rules with some algebra rules that allow us to find the derivative of a large number of functions.

Some basic algebra rules

We have two algebra rules, both very natural, a "sum" rule and a "scalar multiplication" rule.

If we know the derivative of two functions f(x) and g(x), the definition of the derivative shows us that the derivative of

f(x) + g(x)

should just be

f '(x) + g '(x).

Thus the derivative of the sum is the sum of the derivatives.

For example, since the derivative of x^5 is 5x^4 and the derivative of sin x is cos x then the derivative of  x^5+sin x is  5x^4+cos x.

In a similar manner, if we multiply a function f(x) by a constant, c, the derivative of cf(x) is just cf '(x).  Thus the derivative of (10)(x^5) is (10)(5x^4) = 50x^4.

Multiplying a function by a constant is called "scalar multiplication" (we are "scaling" the function) and so the derivative preserves scalar multiplication.

One rule we do not yet have is the derivative of a product of two functions!  That turns out to be more complicated and will be saved for next time.

Other resources

There are some good online applets that show this off.  Let me see if I can find some....