Summary

These techniques are really important, but sometimes they become the focus. It's important to remember that they come right out of the limit definition of the derivative.

For example, the constant rule: if f(x)=k, where k is any real number, then

so

More generally, the power rule: if , where n is any real number, then

In particular, if f(x)=x, f'(x)=1:

A terrifically important rule is the constant multiple of a function rule: if f(x) is a differentiable function (that is, f'(x) exists), and k is any real number, then

Finally, the sum and difference rule is introduced: if f(x)=u(x) + v(x), and if u'(x) and v'(x) exist, then

and if f(x)=u(x) - v(x), then

The derivative of a sum is the sum of the derivatives, whereas the derivative of a difference is the difference of the derivatives.

This rule allows us to use the constant multiple rule, as well as the power rule, to compute the derivative of any polynomial:

Example: #1, p. 223

Example: #7, p. 223

Example: #15, p. 223

Example: #20, p. 223

Example: #32, p. 224

Example: #55, p. 225