4. Applications of Derivatives
Differentials
Problem 4.6.67
Textbook Question
Differentials Consider the following functions and express the relationship between a small change in x and the corresponding change in y in the form dy = f'(x)dx.
f(x) = 3x³ - 4x
Verified step by step guidance1
Identify the function f(x) = 3x³ - 4x and determine its derivative f'(x).
Calculate the derivative f'(x) using the power rule, which states that d/dx[x^n] = n*x^(n-1).
Express the relationship between the small change in x (denoted as dx) and the corresponding change in y (denoted as dy) using the formula dy = f'(x)dx.
Substitute the expression for f'(x) into the equation dy = f'(x)dx to relate dy and dx.
Interpret the result to understand how a small change in x affects the change in y based on the derivative.
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