Post #27

I'm going to continue with more frequently missed questions.

Product Rule
Product rule is used when anything two things are being multiplied together.
Product rule states copy the first times the derivative of the second + copy the second times the derivative of the first or uv' + vu'
Example: Find the derivative of t sin (t)
t (cos t) + sin (t) (1) = t cos t + sin t

Example 2: cos x sin x
cos x (cos x) + (sin x) ( - sin x)
cos ^2 x - sin^2 x

Chain rule
Use chain rule when you are finding the derivative with exponents.
To do chain rule, work from the outside in.
Example: cos^2 (x^2)
It helps me to rewrite it as cos (x^2) ^2
2 (cos (x^2)) (-sin (x^2) (2x)
-4x cos (x^2) sin (x^2)

Ln Integration
This is only used with fractions and when the top is the derivative of the bottom
Example: Integrate 2x / x^2 + 9
The derivative of x^2 + 9 is 2x which is the top of the fraction so the integral is
ln |x^2 + 9 | + c

Example 2: Integrate cos x/ sin x
The derivative of sin x is cos x so
ln |sin x | + c

Equation of a normal line
To find the equation of a normal line, find the perpendicular slope and plug into the point slope formula.
Example: An equation of the line normal to the graph of y= (3x^2 + 2x) ^1/2 at (2,4) is
First step is to take the derivative and plug in x to find the slope.
1/2 (3x^2 + 2x ) ^-1/2 (6x + 2)
6(2) + 2 / 2 (3 (2)^2 + 2(2) )^1/2
m = 14/ 8 or 7/4
Now take the negative reciprocal which is -4/7
Since you are given a point and now have your slope, just plug in.
y-4 = -4/7 (x -2 )
You can simply that until you get an answer choice.

I still need help on numbers like 30 on the calculator portion of the last test we took. It asks how fast is the distance between the boats increasing and also number 44 that says find the distance traveled.