Derivatives

The derivative measures the instantaneous rate of change of a function $f(x)$ with respect to $x$. Defined as the limit of average rates of change, $f^{\prime }(x)={\lim }_{h\to 0}\frac{f(x+h)-f(x)}{h}$, we use standard rules to compute derivatives efficiently without computing limits every time. Fundamental rules include the Power Rule $\frac{d}{dx}[x^n]=n{x}^{n-1}$, the Product Rule $\frac{d}{dx}[uv]=u^{\prime }v+u{v}^{\prime }$, the Quotient Rule $\frac{d}{dx}\left[\frac{u}{v}\right]=\frac{u^{\prime }v-u{v}^{\prime }}{v^2}$, and the Chain Rule $\frac{d}{dx}[f(g(x))]=f^{\prime }(g(x))g^{\prime }(x)$ for composite functions. Combining these rules allows us to differentiate complex expressions involving polynomials, exponentials, logarithms, and trigonometric functions.