Differentiation Rules

Differentiation rules are rules (actually, theorems) used to compute the derivative of a function in calculus. In what follows, all functions are assumed to be differentiable.

Basic Rules

Derivative of a Constant: If $y(x)=c$ is a constant function then $\frac{dy}{dx} = 0$.

Sum Rule: If $y(x) = u(x)+v(x)$ then $\frac{dy}{dx} = \frac{du}{dx} + \frac{dv}{dx}$.

Product Rule: If $y(x) = u(x) \cdot v(x)$ then $\frac{dy}{dx} = u(x)\frac{dv}{dx} + v(x)\frac{du}{dx}$.

Quotient Rule: If $y(x) = \frac{u(x)}{v(x)}$ then $\frac{dy}{dx} = \frac{v(x)\frac{du}{dx} - u(x)\frac{dv}{dx}}{(v(x))^2}$.

Chain Rule: If $y(x) = u(v(x))$ then $\frac{dy}{dx} = \frac{du}{dv}\cdot \frac{dv}{dx}$.

Power Rule: If $y(x) = (u(x))^n$ then $\frac{dy}{dx} = n(u(x))^{n-1} \cdot \frac{du}{dx}$. For integer $n$ this is just a consequence of the product and quotient rules and induction, but it can also be proven for all real numbers $n$, e.g. by using the extended Binomial Theorem.

Derivatives of Trig Functions

Derivative of Sine If $y(x) = \sin x$, then $\frac{dy}{dx} = \cos x$.

Derivative of Cosine If $y(x) = \cos x$, then $\frac{dy}{dx} = -\sin x$.

Derivative of Tangent If $y(x) = \tan x$, then $\frac{dy}{dx} = \sec^2 x$. Note that this follows from the Quotient Rule.

Derivative of Cosec If $y(x) = \csc x$, then $\frac{dy}{dx} = -\csc(x)\cot(x)$.