Trig Inversesap Calculus

Posted on by



Inverse Function Review

One application of the chain rule is to compute the derivative of an inverse function. First, let's review the definition of an inverse function:

  1. AP Calculus-Inverse Trigonometric Functions and Integration.notebook Subject: SMART Board Interactive Whiteboard Notes Keywords: Notes,Whiteboard,Whiteboard Page,Notebook software,Notebook,PDF,SMART,SMART Technologies ULC,SMART Board Interactive Whiteboard Created Date: 5/13/2014 2:40:33 PM.
  2. AP Calculus-Inverse Trigonometric Functions and Differentiation.notebook Subject: SMART Board Interactive Whiteboard Notes Keywords: Notes,Whiteboard,Whiteboard Page,Notebook software,Notebook,PDF,SMART,SMART Technologies ULC,SMART Board Interactive Whiteboard Created Date: 5/13/2014 2:39:40 PM.

Practice the relationship between the derivatives of inverse functions: if f and g are inverse functions, then f’(x)=1/(g’(f(x)). AP Calculus: To Infinity & Beyond! Search this site. College Math Placement. D3 - Trig with a Calculator. D4 - Derivatives using a TI-84+ D5. AP Calculus Implicit Differentiation and Other Derivatives. Answers and explanations. By implicit differentiation, Start by taking the derivative of all four terms, using the chain rule (sort of) for all terms containing a y. Then move all terms containing. To Implicit Differentiation Homework Answers.

We say that the function is invertible on an interval [a, b] if there are no pairsin the interval such thatand . That means there are no two x-values that have the same y-value. That's important, because if two x-coordinates map to the same y-coordinate, the inverse function (working in reverse) would map a single x-coordinate to multiple y-coordinates. That doesn't make sense, because f(x) could have more than one resulting value!

We say thatis the inverse of an invertible functionon [a, b] if:

For example, the functionsandare inverses onsinceon that interval. Note that it works both ways -- the inverse function of the original function returns x, and the original function performed on the inverse ALSO returns x.

Taking The Derivative

So, how do we differentiate an inverse function? Recall the chain rule:


Applying this to the definition of an inverse function, we have:


So:


Let's see how to apply this to real examples.

Example 1

Letsoas above. Then, and applying the formula we have:

This agrees with the answer we would get from viewingas the polynomial function.

Example 2

Trig inversesap calculus formula

The functionis invertible on the interval, with inverse. We know that, so applying our formula we see that


We can check that , which means that

.

Related Pages
Lessons On Trigonometry
Inverse trigonometry
Trigonometric Derivatives
Calculus: Derivatives
Calculus Lessons

Table Of Derivatives Of Inverse Trigonometric Functions

The following table gives the formula for the derivatives of the inverse trigonometric functions. Scroll down the page for more examples and solutions on how to use the formulas.


Example:
Differentiate

Solution:
We can use the above formula and the chain rule.

Example:
Differentiate

Solution:
We use the product rule and chain rule.

Inverse Trigonometric Functions - Derivatives

Inversesap

Formulas for the derivatives of the six inverse trig functions and derivative examples.

Examples:
Find the derivatives of the following functions

  1. f(x) = (sin-1)2
  2. g(t) = cos-1√(2t - 1)
  3. y = tan-1(x/a) + ln√((x-a)/(x+a))
  • Show Video Lesson

Inverse Trigonometric Functions - Derivatives - Harder Example

Example:
Find the derivatives of
y = sec-1√(1 + x2)

Inverse Trigonometric Functions - Derivatives - Harder Example

Example:
Find the derivatives of
y = sin-1(cos x/(1+sinx))

  • Show Video Lesson

Derivatives of Inverse Trig Functions

One example does not require the chain rule and one example requires the chain rule.

Examples:
Find the derivatives of each given function.

  1. f(x) = 3sin-1(x)
  2. g(x) = 4cos-1(3x2)

Trig Inversesap Calculus Worksheet

Derivatives of Inverse Trig Functions

Examples:
Find the derivatives of each given function.

Trig Inversesap Calculus Calculator

  1. f(x) = -2cot-1(x)
  2. g(x) = 5tan-1(2x)
  • Show Video Lesson

Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.


Trig Inversesap Calculus Problems

We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page.