Symbolic Differentiation

Using differentiation rules, we can compute derivatives symbolically — by manipulating formulas rather than computing limits each time. The monomial rule is key: $\frac{d}{dx}(a x^k) = a k x^{k-1}$ for any constants $a, k$.

Combined with the sum rule, this lets us differentiate any polynomial instantly: $\frac{d}{dx}(3x^4 - 2x^2 + 5) = 12x^3 - 4x$. With the chain rule and product rule, we can differentiate any composition of elementary functions.

Symbolic differentiation underlies automatic differentiation in modern ML frameworks: the framework stores the computation graph and applies differentiation rules backward (backpropagation) to compute gradients efficiently.