1. The problem: You asked for a list of first principles formulas and to highlight the important ones. 2. First principles in calculus refer to the definition of derivatives using limits. The fundamental formula is: $$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$ This formula defines the derivative of a function $f$ at a point $x$ as the limit of the average rate of change as $h$ approaches zero. 3. Important related formulas include: - The definition of the derivative of $x^n$: $$\frac{d}{dx} x^n = \lim_{h \to 0} \frac{(x+h)^n - x^n}{h}$$ - The derivative of sine function from first principles: $$\frac{d}{dx} \sin x = \lim_{h \to 0} \frac{\sin(x+h) - \sin x}{h}$$ - The derivative of cosine function from first principles: $$\frac{d}{dx} \cos x = \lim_{h \to 0} \frac{\cos(x+h) - \cos x}{h}$$ 4. These formulas are important because they form the foundation of differential calculus, allowing us to find instantaneous rates of change and slopes of curves. 5. Summary of key first principles formulas: - Derivative definition: $$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$ - Power rule from first principles: $$\frac{d}{dx} x^n = \lim_{h \to 0} \frac{(x+h)^n - x^n}{h}$$ - Sine derivative: $$\frac{d}{dx} \sin x = \lim_{h \to 0} \frac{\sin(x+h) - \sin x}{h}$$ - Cosine derivative: $$\frac{d}{dx} \cos x = \lim_{h \to 0} \frac{\cos(x+h) - \cos x}{h}$$ These are the most important first principles formulas to understand the concept of derivatives.