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)
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 substitute this in.
df/dx = 55(x+1)d/dx(x+1)
df/dx = 55x+55(1)
df/dx = 55x+55
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