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AP Calculus Videos 13 videos

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AP Calculus 1.4 Derivatives 202 Views


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AP Calculus 1.4 Derivatives. Which of the following best describes the quantity?


Transcript

00:00

Thank you We sneak all right Put this problem in

00:05

your calculator in shmoop All right Which of the following

00:08

best describes the quantity d y t ax given the

00:11

relation why cubed minus four x y squared plus eleven

00:14

x cubed equals seven and hear the potential All right

00:19

thinking thinking we're doing okay They're asking us to find

00:24

the y d x which means this is a derivatives

00:27

problem to find the derivative of this relation we typically

00:30

right Why explicitly is a function of x so it

00:33

looks like why equals something But in this case it's

00:37

really hard to isolate Why So we can find the

00:39

derivative implicitly Instead this is called implicit differentiation shockingly which

00:46

basically means the dependent variable Why has not been written

00:50

explicitly in terms of the independent variable x So we

00:54

start by applying the derivative with respect to x to

00:57

each and every term in the equation Well the first

01:00

term is three y squared times d y t ax

01:03

because we're finding the derivative of ah wai term with

01:06

respect to x All right our second term is negative

01:09

for x times Why squared all for this term we

01:13

have to use the product rule because we have two

01:15

terms multiplied by each other Recalled that the product rule

01:18

tells us that the derivative of f of x times

01:22

g of ax equals f of x times the derivative

01:26

of g of x plus the derivative of f of

01:30

x times G of x we can pull out the

01:32

minus four as a constant and we'll get x times

01:36

two why the y d x plus y squared times

01:40

d x d acts which is just one So the

01:43

second term simplifies to negative eight X y c y

01:47

t ax minus for y squared All right Third term

01:51

eleven x cube becomes thirty three x squared We'd multiply

01:55

it by d x d x here too But that's

01:58

still just one and finally the derivative of seven or

02:01

any constant ever is just zero Now it can isolate

02:05

the righty axe and move the terms without d y

02:08

d x to the other side of the equation Well

02:10

the widely axe times three y squared minus eight x

02:13

y equals four y squared minus thirty three x squared

02:17

Then we just divide by three y squared minus eight

02:19

X y to get the idea ax equals four y

02:22

squared minus thirty three x squared all over three y

02:26

squared minus eight x Y look carefully at the answers

02:29

because they all look really similar He is Our answer 00:02:34.0 --> [endTime] is in the river No

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