Rules of Differentiation

Polynomials are easy, but real functions are built by multiplying, dividing, and composing simpler pieces. Three rules — the chain, product, and quotient rules — together with the derivatives of the standard transcendental functions, let us differentiate essentially anything that appears in the IB course.

Derivatives of standard functions. You must know these by heart: $$\frac{d}{dx}(\sin x) = \cos x, \qquad \frac{d}{dx}(\cos x) = -\sin x, \qquad \frac{d}{dx}(\tan x) = \sec^2 x,$$ $$\frac{d}{dx}(e^x) = e^x, \qquad \frac{d}{dx}(\ln x) = \frac{1}{x}.$$ The exponential function $e^x$ is remarkable: it is its own derivative. The natural logarithm is the inverse of $e^x$, and its derivative $\frac{1}{x}$ reappears constantly in integration. Note that these trig derivatives require $x$ to be mea