Ch. 3 - Derivatives

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 3 - Derivatives

Problem 3.10.3

Chapter 3, Problem 3.10.3

Find the slope of the line tangent to the graph of y = tan^−1 x at x= −2.

Verified step by step guidance

First, understand that the slope of the tangent line to the graph of a function at a given point is given by the derivative of the function evaluated at that point.

The function given is y = tan^−1(x), which is the inverse tangent function. We need to find its derivative with respect to x.

The derivative of y = tan^−1(x) is given by the formula:

\frac{1}{x^{2} + 1}

. This formula comes from the differentiation of the inverse tangent function.

Now, substitute x = -2 into the derivative formula to find the slope of the tangent line at this point:

\frac{1}{(^{-2) 2} + 1}

Simplify the expression obtained in the previous step to find the numerical value of the slope. This will give you the slope of the tangent line to the graph at x = -2.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Derivative

The derivative of a function at a given point measures the rate at which the function's value changes as its input changes. It is defined as the limit of the average rate of change of the function over an interval as the interval approaches zero. In this context, finding the slope of the tangent line involves calculating the derivative of the function y = tan^−1(x) at the specific point x = -2.

Tangent Line

A tangent line to a curve at a given point is a straight line that touches the curve at that point without crossing it. The slope of the tangent line represents the instantaneous rate of change of the function at that point. In calculus, the slope of the tangent line can be found using the derivative of the function evaluated at the point of interest.

Inverse Trigonometric Functions

Inverse trigonometric functions, such as tan^−1(x), are the functions that reverse the action of the standard trigonometric functions. For example, tan^−1(x) gives the angle whose tangent is x. Understanding the properties and derivatives of these functions is essential for solving problems involving their slopes and behaviors, particularly when calculating derivatives at specific points.

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Derivatives of Other Inverse Trigonometric Functions

Related Practice

Textbook Question

Find the following higher-order derivatives.

dⁿ/dxⁿ (2^x)

Textbook Question

Evaluate the derivative of the following functions.

f(x) = sin(tan^-1 (ln x))

Textbook Question

Consider the line f(x)=mx+b, where m and b are constants. Show that f′(x)=m for all x. Interpret this result.

Textbook Question

75–86. Logarithmic differentiation Use logarithmic differentiation to evaluate f'(x).

f(x) = (1+ 1/x)^x

Textbook Question

75–86. Logarithmic differentiation Use logarithmic differentiation to evaluate f'(x).

f(x) = (x+1)¹⁰ / (2x-4)⁸

Textbook Question

The function $s(t)$ represents the position of an object at time t moving along a line. Suppose $s(2)=136$ and $s(3)=156$ . Find the average velocity of the object over the interval of time $[2, 3]$ .

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