Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

6. Derivatives of Inverse, Exponential, & Logarithmic Functions

Derivatives of Inverse Trigonometric Functions

Calculus

6. Derivatives of Inverse, Exponential, & Logarithmic Functions

Derivatives of Inverse Trigonometric Functions

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1:40 minutes

Problem 3.10.3

Textbook Question

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.