Less11ImplicDifferentiation

Question 1

What does implicit differentiation allow you to do?

Answer 1

It allows you to find the derivative of y without having a y= function.

y is an implicit function of x, not a direct function.

Question 2

What is the derivative [^d/_dx] of:

xy² ?

Answer 2

Product rule: x*y’ + y*x’

f=x f’=1

g=y² g’=2y

x*2y*y’^{(this is the chain rule)} + y²*1

=x2yy’ + y²

Question 3

What is the drivative of:

x² + y² = 25 ^{(a circle)}

Answer 3

f=x² f’=2x

g=y² g’=2yy’

^{(the derivative of 25 is 0)}

2x + 2yy’ = 0

2yy’ = -2x

y’ = -2x/2y = -x/y

pt. (3,4) = -¾ pt.(-3,-4) = -¾
pt. (-3,4)=¾ pt.(3,-4)=¾

Question 4

Summarize the steps for implicit differentiation.

Answer 4

^{(1) Differentiate both sides with respect to x.}

^{(2) Move all terms involving dy/dx fo the left side and move all other terms to the right side.}

^{(3) Factor dy/dx out of the left side.}

^{(4) Solve for dy/dx}

^{Hint: Because you don’t know what y equals, both y and y’ must remain in the final answer}

Question 5

Find the derivative of the following using the 4 steps:

f(x) = x³ + xy = 7 - y²

Answer 5

^{(1) 3x² + (xy’+y*1)}(productrule)^{=0 - 2y}

^{(2) xy’ + 2yy’ = -3x² - y}

^{(3) y’(x+2y) = -3x² - y}

^{y’ = (-3x² -y)/(x + 2y)}

Question 6

What kind of function is

siny = x and what is its derivative?

Answer 6

This is the inverse sine function.

(1) cosy*y’ = 1
(2) y’ = 1/cosy

Question 7

What are related rates problems?

Answer 7

Both a and y depend on a 3^rd variable, ususally called t.

Over time, x and y change at different rates.

Related rates problems are typically word problems.

Question 8

You drop a pebble in a calm pond and every second the radius increases by 1 foot (^dr/_dt=1 ft). How fast does the area of the circles increase?

What is the rate of change at t=4 and t=8?

Answer 8

A = πr²

^{dA/dt = 2πr*dr/rt}

^{dr/dt=1 :: dA/dt = 2πr}

^{It’s growing at 4 sec = 8π ft²/sec}

^{It’s growing at 8 sec = 16π ft²/sec}

^{pizza principle: a circle grows larger when you add to the outside of the radius}

Question 9

How do you find the first derivative in Implicit Differentiation on the TI nSpire CAS?

Answer 9

^{y= screen → menu → Calculus → Implicit differentiation → type in equation, comma, x, comma, y, enter}

Question 10

How do you find the second derivative in Implicit Differentiation on the TI nSpire CAS?

Answer 10

add a comma then 2 to the end of the first derivative request

Question 11

The second derivative will not be simplified. How do you do that on the TI nSpire?

Answer 11

menu → algebra → #9Fraction tools → #4 Common denominator → ctrl/(-) key gives “Ans” which will use the answer in the above line.

Question 12

Find y’:

x² + y² = 9

Answer 12

f=x² f’=2x

g=y² g’ = 2yy

2x + 2yy’ = 0

2yy’ = -2x

y’ = -2x/2y = =x/y

Question 13

Find y’ by implicit differentiation:

x³ - xy + y² = 7

Answer 13

^{Differentiate both side:}

^{3x² - (xy’ + y)product rule + 2yy’ = 0}

^{xy’ + 2yy’ = -3x² + y}

^{y’(-x + 2y) = -3x² + y}

^{y’ = (-3x²+y)/(-x + 2y)}

Question 14

Find y’ by implicit definition:

tan(x + y) = x

Answer 14

sec²(x+y)(1+y’) = 1

1 + y’ = 1/(sec²(x + y))

y’ = 1/(sec²(x + y)) - 1

y’ = cos²(x + y) - 1

Question 15

Find the second derivative

y”, if y² = x³

Answer 15

y² = x³

2yy’ = 3x²

y’ = 3x²/2y (1st derivative)

^{f=3x² f’=6x g=2y g’=2y’}

^{(2y*6x - 3x²/2y’)/(2y)² quotient rule}

12xy - 3x²*2(3x²/2y) / 4y²

12xy - 3x²(3x²/2y) /2y²

12xy - 9x⁴/2y / 2y²

=6xy - 9x⁴ / 4y³