Tag Archives: Implicit Differentiation

Implicit Differentiation, Tangent Lines, Inverse Trig Functions

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.

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Filed under MAT 131

Derivatives and Implicit Differentiation

How do you find two tangent lines passing through a point? The point is not a point on the line! So decide on an arbitrary point on the line and call it f(a). Then solve for the slope m in terms of a (Find f’(a)). Then plug in your values and solve for the tangent line at the two points.

Derivative Example Problems

Rule: Apply log rules when you see logs!!!!!! (You want to expand as much as possible)

Implicit Differentiation

We briefly started to talk about implicit differentiation. So a function is y=f(x). or “y as described by function f with respect to point x equals”. And this seems to be the key here. So we started using this new notation for derivatives, instead of the usual f prime. To start, look at a very easy example, x2.

y equals x2 which is the same as, f(x) equals x2, which says that y = f(x).”

“The derivative of the function which equals y with respect to point x equals f prime, which is equivalent to the derivative of x2 equal 2x.”

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Filed under MAT 131