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What is the formal definition of a derivative?

The formal definition of derivative of a function y=f(x) is: y’=limΔx→0f(x+Δx)−f(x)Δx. The meaning of this is best understood observing the following diagram: The secant PQ represents the mean rate of change ΔyΔx of your function in the interval between x and x+Δx .


Similarly one may ask, what is the definition of a derivative in calculus?

The definition of the derivative is the slope of a line that lies tangent to the curve at the specific point. The limit of the instantaneous rate of change of the function as the time between measurements decreases to zero is an alternate derivative definition.

Subsequently, question is, what is the limit definition of a derivative? The derivative of function f at x=c is the limit of the slope of the secant line from x=c to x=c+h as h approaches 0. Symbolically, this is the limit of [f(c)-f(c+h)]/h as h→0.

how do you use the formal definition of a derivative?

Formal Definition of the Derivative

  1. Form the difference quotient ΔyΔx=f(x0+Δx)−f(x0)Δx;
  2. Simplify the quotient, canceling Δx if possible;
  3. Find the derivative f′(x0), applying the limit to the quotient. If this limit exists, then we say that the function f(x) is differentiable at x0.

Whats is a derivative?

A derivative is a contract between two or more parties whose value is based on an agreed-upon underlying financial asset (like a security) or set of assets (like an index). Common underlying instruments include bonds, commodities, currencies, interest rates, market indexes, and stocks.

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