Power, Product, and Quotient Rules

Before these rules, finding a derivative required the tedious Limit Definition. The Power Rule is our first major shortcut. It states that for any function $f(x) = x^n$, the derivative is $f'(x) = nx^{n-1}$. This works for positive integers, negative numbers, and even fractions. When a constant is involved, like $cf(x)$, the derivative is simply $c \cdot f'(x)$. This allows us to differentiate long polynomials term-by-term, effectively breaking a complex curve into manageable power-based chunks.

Functions aren't always simple sums; they often multiply or divide. The Product Rule handles $f(x) = u(x)v(x)$ with the formula $f'(x) = u'v + uv'$. Think of it as 'the derivative of the first times the second, plus the first times the derivative of the second.' For division, we use the Quotient Rule: if $f(x) = \frac{u(x)}{v(x)}$, then $f'(x) = \frac{u'v - uv'}{v^2}$. A common mnemonic for the quotient rule is: 'Low d-High minus High d-Low, over Low-Low.'

The derivative of a function is itself a function, meaning we can differentiate it again! The second derivative, denoted $f''(x)$ or $\frac{d^2y}{dx^2}$, measures the rate of change of the slope. In physics, if $s(t)$ is position, then $s'(t)$ is velocity, and $s''(t)$ is acceleration. Understanding higher-order derivatives allows us to analyze not just how fast something is moving, but how that speed is changing over time, which is critical for engineering and physics.