Using this document
I recommend you think for 1-3 minutes after getting stuck before viewing a hint, and 10-15 minutes before viewing giving up and viewing the solution.
Pay attention to the lower and upper bounds! There's no honor in staring blankly at a problem for 30 minutes when you're stuck, and likewise viewing solutions prematurely is inefficient.
Inverse and Implicit Differentiation
Using the fact that and find
Differentiate both sides of then solve for
Taking the derivative of both sides and using the chain rule gives
Since dividing by isolates to give
Derive the product rule using the chain rule and logarithms.
Logarithms turn multiplication into addition, and we know how to handle addition (derivative of sum is sum of derivatives!)
Let (or more verbosely ) then take the log of both sides
Differentiate both sides, the chain rule and the fact that imply giving
Since this becomes
Solving for by multiplying both sides by finally gives the familiar product rule
Generalize the product rule to a product of more than two functions.
Use the logarithm approach from the previous exercise.
Let , take logs to get
Multiply both sides by and substitute back in for
Compute the following derivative
and take the logarithm of both sides
Take the log of both sides and simplify
Take derivatives to get
Multiply by and substitute back in
In probability, we need to compute the following sum to find the expected value of a random variable with a geometric distribution
Using the identity find the sum above
Differentiate both sides of the identity with respect to
Multiply both sides by to turn into , multiply by to get the desired sum
(Footnote about probability: since the expectation simplifies to .)
The pattern above was that applying any linear operation to both sides of an identity involving a sum gives a new identity. Using this trick find the sum of