Implicit Differentiation, Tangent Lines, Inverse Trig Functions

Published on: Tue Mar 16 2010

Find the tangent line at point (3,2) Now that we know dy/dx, solve for the slope using the given values for (x and y) Now use the Tangent Line formula Next Problem Now to find the tangent line for this problem, with the coordinates (2,0) A brief aside Prove derivative of ln(x) using Implicit Differentiation
  1. The Problem
  2. Write in full form, including y
  3. Use log properties to arrange into x = form
  4. Take the derivative with respect to the change in x from both sides
  5. The derivative of “e to the y” remains “e to the y”, the derivative of x is one.
  6. Rearrange to solve for the derivative dy/dx.
  7. Substitute for y.
  8. “e to the ln of anything” is equal to the anything, in this case anything is x.
Find the derivative of arcsin x.
  1. The Problem
  2. Rearrange the problem using Trig
  3. Take the derivative from both sides
  4. x is equal to 1
  5. Solve for dy/dx
  6. Substitute for y in the solution.
  7. The professor kept going and somehow (using Trig?) managed to bring the answer to this.
This is what he did: Not quite sure how cos y equaled x Increasing and decreasing functions For sin to be invertable, you need to restrict the domain to -pi/2 to pi/2 To see how a function looks, determine the invervals above and below zero. To do this solve for x. You can look at the derivaties to decide in a function is concave up or concave down. Second derivative is negative, concave down. Second derivative is positive, concave up READ problem 2.7 #43 from the textbook.