Differentiating simple functions

A function f, is a rule that assigns a single value of the dependent variable y, (in the codomain) to each value of the independent variable x in the domain of the function, such that y = f(x)

The rate of change, or derivative of a function f(x) at x = a, can be interpreted as the gradient of the graph of the function at the point x = a.

The derivative is defined more formally in terms of a limit, and may be represented in a variety of ways. If y = f(x)

f′(x) = dy/dx = lim x →0)(Δyx) = lim( h→0) {[f(x + h) – f(x)]/h}

The sums, constant multiples, product and quotient rules for differentiating combinations of functions are;

Sum rule: (f + g)′ = f′ + g

Constant multiple rule: (kf)′ = kf

Product rule: (fg)′ = fg + fg

Quotient rule: (f/g)′ = (fgfg′)/g2

Reciprocal rule: 1/f′ = – f′/f2

Standard derivatives
f(x) f′(x)
k (constant) 0
kxn nkxn-1
sin kx k cos kx
cos kx k sin kx
tan kx k seckx
cosec kx k cosec kx cot kx
sec kx k sec kx tan kx
cot kx k coseckx
exp(kx) exp(kx)
loge(kx) 1/x

The derivatives of ax and loge x are given by

d/dx(ax) = (loga)ax


d/dx(loga x) = 1/(x loga)

Logarithms to different bases are related by

logb = (logc)/(logb)


~ by jamesdow2013 on April 19, 2013.

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