Page 27 - Calc for the Clueless

We now solve for dy/dx. Once we take the derivative, it becomes an elementary algebra equation in which we
solve for dy/dx.

1. All the terms without dy/dx go to the other (right) side and change signs.

2. All terms without dy/dx on the other side stay there; no sign change.

3. All terms with dy/dx on the right go to the left and change signs.

4. All terms with dy/dx on the left stay there; no sign change.

5. Factor out dy/dx from all terms on the left; this coefficient is divided on both sides. Therefore it goes to the
bottom of the fraction on the right.

6. Rearrange all terms so that the number is first and each letter occurs alphabetically.

It really is easy with a little practice. Using this method, our answer is

Example 16 Continued—

Maybe you think this is too complicated. Let's do another with much simpler coefficients after taking the
derivative: Ay' + B - Cy' - D = 0.

B and D have no y' and must go to the other side and flip signs. The (A - C) is factored out from the y' and goes
to the bottom of the answer. So y' = (D - B)/(A - C).

Still don't believe? Let's do it step by step.

So

If you look closely, this really is Example 14, the algebraic part.

Example 16 Last Continuation—

Suppose we are given the same equation, but are asked to find dy/dx at the point (-2,1).

The first step is the same.

But instead of doing all the rest of the work, we substitute x = -2 and y = 1!