Find the equation of the tangent line to the curve at the given x -value. y=(x^2-1)^3 ; x=2

Name: Problem 26, Chapter 2: Applied Calculus For Business, Economics, and Finance
Uploaded: 2021-02-03T20:35:34-08:00
Duration: 2 min 37 s
Channel: Gregory Higby
Description: Find the equation of the tangent line to the curve at the given x -value. y=(x^2-1)^3 ; x=2

step 1 The overall theme of a tangent line problem is to make sure that you have a point and a slope. W

Find the equation of the tangent line to the curve at the...

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Key Concepts

Chain Rule

The chain rule is a method for differentiating composite functions. It allows one to find the derivative of a function that is composed of an outer function and an inner function by differentiating the outer function with respect to the inner function and then multiplying by the derivative of the inner function.

Tangent Line

A tangent line to a curve at a given point is a straight line that touches the curve at that point and has the same instantaneous slope as the curve. It represents the best linear approximation of the curve near that point and is fundamental in understanding the local behavior of functions.

Derivative

The derivative of a function provides the slope of the tangent line at any point on the curve. It is defined as the limit of the difference quotient as the interval approaches zero, and serves as a key tool in determining rates of change and understanding dynamic behavior in functions.

Evaluation at a Specific Point

Evaluating a function and its derivative at a specific point involves substituting the given x-value into the function and its derivative. This step is crucial for determining the exact coordinates of the point of tangency and the precise slope of the tangent line at that point.

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