At this point, you may be wondering why you can’t use the slope formula from algebra to find the slope. The answer is: you still can. There’s no difference between finding a slope using the slope formula, or finding the slope using differentiation. The major difference is that differentiation can give you all of the slopes at all of the points, while the slope formula can only give you one at a time. Let’s say you had a thousand coordinate points for the position of a new planet. You can either plug those in a thousand times to the slope formula or you can differentiate once. It’s faster, easier, and a lot less tedious.
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It’s not uncommon to get to the end of a semester and find that you still really don’t know exactly what one is! That’s because the definition isn’t immediately intuitive; you really get to grasp what one is after you’ve practiced—and practiced. It’s like knowing what an embouchure is in clarinet playing; you can be told that it’s a tongue placement, but it takes many weeks (sometimes months) of practice before you really get a good grasp of the perfect enbouchere and why it’s important.
What Does Differentiate Mean?
- Multiply the top and the bottom by √( 4( x + Δx) + 1 + √(4x + 1):
f'(x) = lim Δx → 0 √( 4( x + Δx) + 1 – √(4x + 1) ) * √( 4( x + Δx) + 1 + √(4x + 1) / Δx* √( 4( x + Δx) + 1 + √(4x + 1)
which reduces to:
= lim Δx → 0 4(x + Δx) + 1 – (4x + 1) / Δx(√ (4x + Δx) + 1) + √ 4x + 1
- Distribute the 4:
= lim Δx → 0 (4x + 4Δx + 1 – 4x – 1) / (Δx(√ (4x + Δx) + 1) + √ (4x + 1)
- Delete terms. In this case you can delete 4x, Δx and 1.
= lim Δx → 0 4 / ((√ (4x + Δx) + 1) + √ 4x + 1))
Step 2: Close the parentheses “)”, then type a vertical bar (called the “with” symbol). On the TI-89, you’ll find the “|” key on the left hand side. Don’t press enter yet.
