I don't understand derivatives at all. My exam is in 3 days and I'm panicking.
No need to panic — 3 days is enough to get a solid understanding. Let me break it down simply:
A derivative tells you the RATE OF CHANGE of something. Think of it like speed — if position changes over time, the derivative is how fast it's changing at any exact moment.
Let's start with the basics. Do you know what f(x) = x² looks like on a graph?
Yeah, it's a parabola. The U-shaped curve.
Exactly! Now here's the key insight:
At x = 0 (bottom of the U), the curve is flat → rate of change = 0
At x = 2, it's going up steeply → rate of change is high
At x = -2, it's going down steeply → rate of change is negative
The derivative f'(x) = 2x gives you this rate at ANY point. Plug in x = 0 → you get 0 (flat). Plug in x = 2 → you get 4 (steep upward). Plug in x = -2 → you get -4 (steep downward).
Makes sense so far? Want to try one together?
Oh, that actually clicks. So the derivative of x³ would be 3x²?
You just discovered the power rule! For any f(x) = xⁿ, the derivative is f'(x) = n·xⁿ⁻¹.
So yes: x³ → 3x², x⁴ → 4x³, x⁵ → 5x⁴.
Now let's level up. What do you think the derivative of 3x² + 5x + 2 would be? (Hint: take each term separately.)
6x + 5? The constant 2 disappears because it's not changing?
That's exactly right — and the reasoning is perfect. Constants don't change, so their rate of change is zero.
You've just learned 80% of what you need for basic derivative questions. Here's your 3-day study plan:
Day 1 (today): Power rule + constant rule + sum rule (we just did these)
Day 2: Product rule and chain rule
Day 3: Practice problems + review
Want to do 5 practice problems right now to lock this in?