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 (Δx →0)(Δy/Δx) = 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)′ = f′g + fg′
Quotient rule: (f/g)′ = (f′g – fg′)/g^{2 }
Reciprocal rule: 1/f′ = – f′/f^{2}
Standard derivatives | |
f(x) | f′(x) |
k (constant) | 0 |
kx^{n} | nkx^{n-1} |
sin kx | k cos kx |
cos kx | –k sin kx |
tan kx | k sec^{2 }kx |
cosec kx | –k cosec kx cot kx |
sec kx | k sec kx tan kx |
cot kx | –k cosec^{2 }kx |
exp(kx) | k exp(kx) |
log_{e}(kx) | 1/x |
The derivatives of a^{x} and log_{e }x are given by
d/dx(a^{x}) = (log_{e }a)a^{x}
and
d/dx(log_{a} x) = 1/(x log_{e }a)
Logarithms to different bases are related by
log_{a }b = (log_{a }c)/(log_{c }b)