Computing derivatives directly from the limit definition is tedious, so a collection of rules streamlines the process. The Power Rule states that if \(f(x) = x^n\) for any real exponent \(n\), then \(f'(x) = n x^{n-1}\). This rule applies to integer, fractional, and negative exponents alike, making it one of the most frequently used results in differentiation. For products of functions, the Product Rule applies: if \(f(x) = u(x) \cdot v(x)\), then \(f'(x) = u'(x) v(x) + u(x) v'(x)\). The Quotient Rule handles ratios: if \(f(x) = u(x)/v(x)\), then \[ f'(x) = \frac{u'(x) v(x) - u(x) v'(x)}{[v(x)]^2}. \]
The Chain Rule governs compositions of functions. If \(y = f(g(x))\), then \(\frac{dy}{dx} = f'(g(x)) \cdot g'(x)\), or equivalently in Leibniz notation, \(\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}\). The Chain Rule is essential whenever one function is applied inside another, and it underlies many derivative formulas. For example, to differentiate \(f(x) = a^x\) for a positive constant \(a\), one rewrites \(a^x = e^{x \ln a}\) and applies the Chain Rule, yielding \(f'(x) = a^x \ln a\). When \(a = e\), this gives the elegant result \(f'(x) = e^x\), showing that the exponential function is its own derivative.
Trigonometric derivatives are immediate consequences of the basic rules. The derivative of \(\sin x\) is \(\cos x\), derivable from the limit definition together with the fundamental limits \(\lim_{h \to 0} \frac{\sin h}{h} = 1\) and \(\lim_{h \to 0} \frac{\cos h - 1}{h} = 0\). Differentiating \(\cos x\) gives \(-\sin x\), while \(\tan x = \frac{\sin x}{\cos x}\) has derivative \(\sec^2 x\) by the Quotient Rule. For the natural logarithm, \(\frac{d}{dx} \ln x = \frac{1}{x}\) for \(x > 0\), a result that follows from the fact that \(\ln x\) is the inverse of \(e^x\) and can be derived using implicit differentiation. The derivative of \(\arcsin x\) is \(\frac{1}{\sqrt{1 - x^2}}\) for \(|x| < 1\), obtained through implicit differentiation of the relation \(\sin y = x\).