1. The problem is to create a cheat sheet for derivatives to help with a math test. 2. The derivative of a function $f(x)$ is defined as $$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$ which measures the rate of change of the function. 3. Important derivative rules: - Power Rule: $$\frac{d}{dx} x^n = n x^{n-1}$$ - Constant Rule: $$\frac{d}{dx} c = 0$$ where $c$ is a constant - Constant Multiple Rule: $$\frac{d}{dx} [c f(x)] = c f'(x)$$ - Sum Rule: $$\frac{d}{dx} [f(x) + g(x)] = f'(x) + g'(x)$$ - Product Rule: $$\frac{d}{dx} [f(x) g(x)] = f'(x) g(x) + f(x) g'(x)$$ - Quotient Rule: $$\frac{d}{dx} \left[ \frac{f(x)}{g(x)} \right] = \frac{f'(x) g(x) - f(x) g'(x)}{[g(x)]^2}$$ - Chain Rule: $$\frac{d}{dx} f(g(x)) = f'(g(x)) g'(x)$$ 4. Common derivatives: - $$\frac{d}{dx} e^x = e^x$$ - $$\frac{d}{dx} \ln x = \frac{1}{x}$$ - $$\frac{d}{dx} \sin x = \cos x$$ - $$\frac{d}{dx} \cos x = -\sin x$$ - $$\frac{d}{dx} \tan x = \sec^2 x$$ 5. Use these rules to find derivatives quickly and accurately on your test.