6. Write the equation of the tangent line at a point. 1.Find the point: plug x into the original equation. 2.Take the derivative 2.Find the slope: evaluate the derivative at the given x. 4. Write the equation: y-y 1 =m(x-x 1 )

7. Write the equation of the tangent line at x=1. 1. 2. 3. 4.

8. A normal line is perpendicular to the graph of a function Remember: perpendicular slopes are negative inverse of each other.

9.  In words, the Product Rule says that “the first function times the derivative of the second, plus the second function times the derivative of the first.”

10. The quotient rule is best summed up with a ditty: Low d-high minus high d-low EIEIO Draw a line, low squared below EIEIO with Low d-high here and high d-low there Low d-high, high d-low, underneath a (low)(low) Old MacGuseman taught a song with the quotient rule!

11.  Position: usually s(t), a function of time  Velocity: v(t), which is (direction matters + or - )  Speed is absolute value of velocity. ( no direction)  Acceleration: a(t), which is  An object stops when velocity is 0.

12. Velocity++--Acceleration+-+- Speeding up Slowdown S SS Speeding up Slowdown S SS Same signs: speeding up Different signs: slowing down

13. Memorize these! No really – Memorize them!

14. (these are needed for the proofs of the trig derivatives that we will do later)

15. Remember this by saying: “Derivative of the outside (inside does not change) Times the derivative of the inside”

16.  Differentiate 1. y = sin(x 2 ) 2. Rewrite into separate functions: d. dx d. du d. dx

17. 1. Start on one side and differentiate each term in the equation with respect to x ; “think chain rule” 2. Then solve the resulting equation for y.

18.  For the circle x 2 + y 2 = 25, find a) dy/dx b) an equation of the tangent at the point (3, 4). b) At the point (3, 4) we have x = 3 and y = 4, so dy/dx = –3/4

20.  This leads to the following formula:  In a similar way we can show

21.  This leads to the following formula: As a special case we take a = e, so ln a = 1 :

22.  For example, we find (d/dx) ln(sin x) :

23.  Use this technique to differentiate in two scenarios: 1. HAVE TO use it when there is a variable in the exponent. 2. Might use it when the function is very complicated. ex.

24.  In general: 1. Take natural log of both sides 2. Use log rules to simplify 3. Differentiate implicitly with respect to x. 4. Solve the resulting equation for y.

26.  CH 3 at a glance 17 days! I will take it, I will pass!