Differentiation from First Principles & Power Rule 📐

### Welcome to Further Mathematics, Scholar! 📐 Calculus is the mathematical engine of aerospace engineering, quantitative finance, robotics, and artificial intelligence! **1. The Gradient of a Curve**: Unlike a straight line with a constant slope $m$, the gradient of a curve changes at every single point! To find the slope of the tangent at point $(x, f(x))$: $$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$ **2. First Principles Proof for $f(x) = x^2$**: $$f(x+h) = (x+h)^2 = x^2 + 2xh + h^2$$ $$\frac{f(x+h) - f(x)}{h} = \frac{x^2 + 2xh + h^2 - x^2}{h} = \frac{2xh + h^2}{h} = 2x + h$$ Taking the limit as $h \to 0$: $$f'(x) = 2x$$ **3. The General Power Rule**: For any real power $n$: $$\frac{d}{dx}(a x^n) = a \cdot n x^{n-1}$$ * If $y = 5x^4 \implies \frac{dy}{dx} = 5 \times 4 x^3 = 20x^3$!