chain rule
in calculus, the chain rule is used to differentiate compositions of functions. it states that for any function f which is dependent on a variable u, and u is a function of a second variable x, then f is a function of x.
in set theory:
f(u)=u v u(x)=x v vf(x)
in calculus:
df/dx = du/dx(dx/du)
to differentiate (x+1)^2, one could multiply this out and apply the sum rule (derivative of a sum is the sum of the derivatives); but what if it were (x+1)^55? suddenly the multipling out doesn’t look so nice. to apply the chain rule, we must first define the functions. let f(x)=(x+1)^55 and u=(x+1). therefore f(x)=u^55. by the chain rule:
df/dx = df/du(du/dx)
df/dx = d/du(u^55)(du/dx)
df/dx = 55u(du/dx)
we earlier defined the variable “u” as (x+1). now we subst-tute this in.
df/dx = 55(x+1)d/dx(x+1)
df/dx = 55x+55(1)
df/dx = 55x+55
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