Differentiating is the act of finding the derivative of a function, thus allowing you to find out how the function changes as its input changes, such as finding the rate of change of the gradient of the function, and when differentiating with respect to time can allow you to give equations explaining the motion of various objects, depending on how you differentiate the functions. There are two main types of differentiation, ordinary differentiation and partial differentiation, the rules outlined below are for ordinary differentiation. While the rules for partial differentiation are not that dissimilar, they do not need to be known outside of university level mathematics and physics.

Rules of (Ordinary) Differentiation

(using f' and g' to denote the derivative of the functions f and g of x respectively, x is a variable, o indicates a composite function, all other letters are constants, rules in bold are important)

Elementary rules

f = xn, f' = nx(n-1) elementary power rule

f = a, f' = 0 constant rule

f = ax, f' = a derivative of a linear function is a constant

(af +bg)' = af'+bg' linearity of differentiation, leading to the 3 following,

(af)' = af' constant multiple rule

(f+g)' = f'+g' sum rule

(f-g)' = f'-g' subtraction rule,

(fg)' = f'g + fg' product rule

(fog)' = (f(g))' = (f'(g))g' = (f'og)g' chain rule

f = 1/g, f' = -g'/g2 reciprocal rule

(f/g)' = (f'g-fg')/g2 quotient rule

Rules for trigonometric functions

f=sin(x), f'=cos(x)

f=cos(x), f'=-sin(x)

f=tan(x), f'=sec2(x)

f=sec(x), f'=sec(x)tan(x)

f=cosec(x) f'=-cosec(x)cot(x)

f=cot(x), f'=-cosec2(x)

Rules for exponential and logarithmic functions (log representing natural logs)

f=exp(ax), f'=a*exp(ax)

f=exp(axn), f'=anx(n-1)*exp(axn)

f=ax, f'=(log(a))*ax

f=log(x), f'=1/x

f=log(xn), f'=nx(n-1)/xn

f=xx, f'=xx(1+log(x))

these are just the simple rules of differentiation for various functions, there are a LOT more, but they are generally only of use at university levels.

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