Formal proof of chain rule

Reference no: EM13127772

Theorem: Let X, Y be subsets of R, let x_0 belong to X be a limit point of X, and let y_0 belong to Y be a limit point of Y. Let f:X-->Y be a function such that f(x_0)=y_0, and such that f is differentiable at x_0. Suppose that g: Y-->R is a function which is differentiable at y_0. Then the function g o f: X-->R is differentiable at x_0, and (g o f)' (x_0) = g'(y_0)f '(x_0)

Reference no: EM13127772

Questions Cloud

Probability that each four players is dealt exactly one ace : You read in a book about bridge that the probability that each of the four players is dealt exactly one ace is about 0.11. This means that in every 100 bridge deals each player has one ace exactly 11 times.

What are the roles women in islam : What are the roles women in Islam? What kind of variation is there from country to country?

Weighted average rate for the year : Darron Co. was formed on January 1, 2011 as a wholly owned foreign subsidiary of a U.S. corporation. Darron's functional currency was the stickle (§). The following transactions and events occurred during 2011:

What is the new volume : The volume of a gas is 5.0L when the temperature is 5.0C. If the temperature is increased to 10.0C without changing the pressure what is the new volume?

Formal proof of chain rule : Suppose that g: Y-->R is a function which is differentiable at y_0. Then the function g o f: X-->R is differentiable at x_0, and (g o f)' (x_0) = g'(y_0)f '(x_0)

What is the molecularity of elementary reaction : What is the molecularity of the following elementary reaction? NH2Cl(aq) + OH¯(aq) -> NHCl¯(aq) + H2O(l)

Maximal and minimal by derivatives : A homeowner wants to enclose a 5,800 square feet rectangular garden by a fence in his backyard. If three sides of the fence cost $6.25 per foot and 4th side costs $10.25 per foot,

Determining the income tax payable : The estimated litigation expense of $1,000,000 will be deductible in 2011 when it is expected to be paid. Use of the depreciable assets will result in taxable amounts of $500,000 in each of the next three years. The income tax rate is 30% for all ..

How many feet of copper can be obtained from the ingot : If the copper is drawn into wire whose diameter is 8.50 mm , how many feet of copper can be obtained from the ingot? The density of copper is 8.94 g/cm^3 .

User Account

All Pages