Page 29 - Calc for the Clueless

Example 18—

the derivative of the sine of a crazy angle is the cosine of that crazy angle times the derivative of the crazy
angle— implicitly and with the product rule the product rule; the minus sign in front means both parts of the
product are negative.

Solving for dy/dx in one step, we get

The only difference in the algebra is to multiply out the terms in parentheses in your head (the cosine mul tiplies
each term). The example is then virtually the same as Example 14.

Antiderivatives and Definite Integrals

We are interested in the antiderivative. That is, given a function f(x), the antiderivative of f(x), F(x), is a
function such that F'(x) = f(x). We are going to explore methods of getting F(x).

The big problem in antiderivatives is that there is no product rule and no quotient rule. You might say, ''Hooray!
No rules to remember!" In fact, this makes antiderivatives much more difficult to find, and for many functions
we are unable to take the antiderivatives. However, in calculus I, antiderivatives are very gentle. Only later do
they get longer and more difficult.

Rule 1

If F'(x) = G'(x), then F(x) = G(x) + C. If the derivatives are equal, the original functions differ by a constant. In
other words, if you have one antiderivative, you have them all; you just have to add a constant.

Example 19, Sort of—

F'(x) = 2x = G'(x). The difference between F(x) and G(x) is a constant, 10.

Rule 2

Add one to the original exponent and divide by the new exponent plus a constant.

If f'(x) = x with , then
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