Unit 3 - Differentiation: Composite. Implicit, and Inverse Functions

Question 1

Composite function

Answer 1

Function inside another function

Question 2

Chain Rule is used to find the

Answer 2

Derivative of composite functions

Question 3

d/dx f(g(x)) =

Answer 3

f’(g(x)) * g’(x)

Question 4

Explicit equation

Answer 4

Equation that is set to be equivalent to y

Question 5

Implicit euqation

Answer 5

Equation that is not set to be equal to y

Question 6

d/dx x =

Answer 6

1

Question 7

d/dx y

Answer 7

dy/dx

Question 8

To implicitly differentiate,

Answer 8

1 - take derivative normally, each time a ‘y’ is involved, include a dy/dx
2 - gather all terms with dy/dx on left side, everything else on right
3 - factor out dy/dx if necessary to create only one dy/dx term
4 - solve for dy/dx

Question 9

To find slope with an implicit derivative,

Answer 9

you need both x and y values

Question 10

To find equation of a tangent line for an implicitly differentiable equation, you need to

Answer 10

1 - find y and x values
2 - find the derivative using implicit differentiation
3 - use derivative to find slope at points
4 - substitute values into y -y1 = m (x-x1)

Question 11

Horizontal tangent

Answer 11

Exists when slope is equal to zero

Question 12

To calculate horizontal tangent for an implicit equation

Answer 12

1 - take derivative and set equal to zero
2 - numerator is equal to zero and solve
3 - substitute x-value into original equation to find y-value

Question 13

Vertical tangent

Answer 13

Exists when slope is undefined

Question 14

To calculate vertical tangent for an implict equation

Answer 14

1 - take derivative and set equal to undefined
2 - denominator is equal to zero and solve
3- substitute y-value into original equation to find x-value

Question 15

Inverse function

Answer 15

Can be found by swapping input and output

Question 16

Three ways to say that f and g are inverses

Answer 16

f is the inverse of g
g(x) = f-1(x)
f(g(x)) = x and g(f(x)) = x

Question 17

d/dx [f-1(x)] =

Answer 17

1 / f’ * [f-1(x)]

Question 18

Inverse trig function notation

Answer 18

notation with a “-1” as in “sin-1(x)”

with prefix “arc” as in arcsin

Question 19

d/dx sin-1(x) =

Answer 19

1 / square root of (1 - x^2)

Question 20

d/dx cos-1(x) =

Answer 20

• 1 / square root of (1 - x^2)

Question 21

d/dx sec-1(x) =

Answer 21

1 / |x| square root of (x^2 - 1)

Question 22

d/dx csc-1(x) =

Answer 22

• 1 / |x| square root of (x^2 - 1)

Question 23

d/dx tan-1(x) =

Answer 23

1 / x^2 + 1

Question 24

d/dx cot-1(x) =

Answer 24

• 1 / x^2 + 1

Question 25

For inverse trig derivatives,

Answer 25

Use derivative multiplied by derivative of value in parentheses

Question 26

In order to simplify inverse trig derivatives,

Answer 26

1- determine whether den/nom is pos/neg
2- simply num with den and leave square root alone
3- determine if you need an absolute value

Question 27

For inverse trig function, the domain and range

Answer 27

of their normal trig function are switched

Question 28

Higher order derivatives notation

Answer 28

y^(n) or d^ny / dx^n

Question 29

Second derivative

Answer 29

Derivative of the derivative

Question 30

When calculating second der. with implicit diff.,

Answer 30

A dy/dx with remain, then you substitute