Wrote the equation of a tangent line for a given function that required using the product rule. Wrote the equation of a normal line (i.e. perpendicular to the curve) for a given function that required using the quotient rule. Found the derivative of tangent by using a trig identity and the definition of secant along with the quotient rule. Used patterns to verify other trig derivatives, so now we have all 6 in our toolkit.
Notes from board
Homework
p. 105 #65-68 (don't use power rule, use alt. form of derivative), #85-89
p. 125 #3-33 (multiples of 3), 63-66, 81, 82
feel free to utilize calcchat.com to help
Resources Similar to today's class
tangent line, function requires product rule: link
normal line, doesn't use quotient rule but otherwise similar: link
using the quotient rule: link
derivative of tangent, explained: link
derivative of secant, explained: link
Resources for Homework
using alt. form of derivative to find derivative at point (#65-68) link starts at 14:20
determining if a function is differentiable at a point (#85-89): link (you can use power rule)
write the equation of a tangent line (#63-66) link (made by me:))
product and quotient rule from graphs (#81-82): link
Went over homework, then found the equation of a tangent line involving quotient rule and the new tan(x) derivative. Looked at 4 functions to see what made each one unique and found one whose derivative we didn't know how to find (a composite function, a function whose 'argument' is not just x). Noticed graphically that the derivative was not straightforward and then reviewed composite functions and found the derivative using algebraic methods and the power rule. Used this to reverse-engineer the chain rule, which says to take the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function. Worked through several different examples of using the chain rule.
Notes from board
Homework
this handout, pick any ten from #1-12
Resources
simple chain rule examples: link
more chain rule examples: link
proof of chain rule, for those interested: link
Warm up involved using the chain rule in tandem with the product rule as well as using the chain rule multiple times in the same problem. Went over homework (detailed answers here) then looked back at our history with all the different function families and whether or not we could take the derivative of them. Noticed we can't take the derivative of exponential and log functions yet, so reviewed their properties as well as how to define the number e as a limit at infinity. Went through the long proof of how the derivative of ln(x) is actually 1/x and how this implies that e^x is its own derivative (here are both proofs as a single video). Worked through a few examples of exponential and log derivatives that also made use of the chain rule.
Notes from board
Homework
this handout, any 3 from 556-573, any 3 from 574-589, any 3 from 596-610
assessment will be Friday 10/27
Resources
some examples using log and exponent derivatives (made by me!!) link
product and chain rule, combined with exp derivative (made by me!!) first half of this
more examples link1 link2 link3 link4