Find the value of the derivative (if it exists) at each indicated extremum. f(x)=(x+2)^2 / 3

Name: Problem 5, Chapter 4: Calculus Early Transcendental Functions
Uploaded: 2021-07-17T00:29:31-08:00
Duration: 1 min 56 s
Channel: Jacquelyn Trost
Description: Find the value of the derivative (if it exists) at each indicated extremum. f(x)=(x+2)^2 / 3

step 1 For this problem, we've been given a function, F of X equals x plus 2 raised to the 2 thirds pow

Find the value of the derivative (if it exists) at each...

Key Concepts

Differentiability at Points with Fractional Exponents

When dealing with functions that include fractional exponents, differentiability issues may arise at specific points, especially at the boundaries of the function's domain. Checking the limits from both sides helps in determining whether the derivative exists at those points or if there is a cusp, corner, or vertical tangent.

Power Rule

The power rule offers a straightforward method for differentiating functions of the form x^n, where the derivative is given by n*x^(n-1). This rule is applicable even when the exponent is fractional, which simplifies the differentiation process for power functions.

Critical Points and Extrema

Critical points are points at which a function's derivative is zero or fails to exist. These points are significant because they may correspond to local extrema (maxima or minima), and analyzing the derivative at these points helps determine the function's behavior and shape.

Derivative

A derivative represents the instantaneous rate of change of a function with respect to its variable. It is calculated using the limit of the difference quotient, providing insights into the local behavior of functions such as slopes of tangent lines at a point.

Chain Rule

The chain rule is a fundamental differentiation technique used for composite functions. When a function consists of an inner function and an outer function, its derivative is found by multiplying the derivative of the outer function (evaluated at the inner function) by the derivative of the inner function.

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