Given f(x)=x^3, x ≥0 =x^2, x<0 Find f^'(0)

Name: Problem 19, Chapter 4: Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry
Uploaded: 2022-05-26T11:47:47-08:00
Duration: 7 min 29 s
Channel: Souvik Ghosh
Description: Given f(x)=x^3, x ≥0 =x^2, x<0 Find f^'(0)

step 1 The function given here is fx equals s cube for all x greater than equals to 0 and equal to x sq

Given f(x)=x^3, x ≥0 =x^2, x<0 Find ...

Key Concepts

Limit Definition of Derivative

The limit definition of the derivative expresses the derivative of a function at a point as the limit of the difference quotient as the interval approaches zero. This foundational concept in calculus provides the framework for analyzing the instantaneous rate of change of the function. It is particularly important when working with piecewise functions, where ensuring that this limit exists from both directions is necessary for differentiability.

Differentiability

Differentiability refers to the existence of a derivative at a given point, meaning the function has a well-defined tangent at that point. For functions, especially those defined piecewise, checking differentiability involves verifying that the limit of the difference quotient exists and is the same from both sides. This concept encapsulates the idea that a function's graph should have no sharp corners or cusps at the point in question.

Piecewise Functions

Piecewise functions are functions that have different expressions or rules for different intervals of the domain. Understanding piecewise functions is crucial when analyzing their properties, particularly at the boundaries where the expressions change. This concept is essential when assessing continuity and differentiability, as these properties may depend on how the separate pieces connect at a transition point.

One-Sided Limits

One-sided limits involve evaluating the limit of a function as the independent variable approaches a specific value from one side only—either from the right (positive side) or the left (negative side). This concept is especially critical when dealing with piecewise functions, as the behavior of the function may differ depending on the direction of approach, affecting the existence and computation of derivatives at transition points.

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