Unit 3: Differentiation: Composite, Implicit, and Inverse Functions Flashcards
What is the derivative of a composite function?
The derivative of a composite function is found using the chain rule.
Fill in the blank: The chain rule is used to differentiate __________ functions.
composite
What is the general formula for the chain rule?
If y = f(g(x)), then dy/dx = f’(g(x)) * g’(x).
What is implicit differentiation?
Implicit differentiation is a method used to find the derivative of functions with x’s and y’s scrambled.
True or False: Implicit differentiation requires that y be isolated on one side of the equation.
False
When using implicit differentiation, what must you remember to apply when differentiating y?
You must apply the chain rule and y’
Derivative of the inverse function
If g is the incerse of f, then g’(x) = 1/f’(g(x))
True or False: If f(x) is a one-to-one function, then it has an inverse function.
True
What is the relationship between a function and its inverse regarding their derivatives?
The derivative of the inverse function is the reciprocal of the derivative of the function
What is the first step in finding the derivative of a composite function?
Identify the outer and inner functions.
What does the notation dy/dx represent?
It represents the derivative of y with respect to x.
True or False: The derivative of a constant function is zero.
True
Fill in the blank: The derivative of ln(x) is __________.
1/x
If h(x) = u(x)v(x), what is h’(x) using the product rule?
h’(x) = u’(x)v(x) + u(x)v’(x).
What is the formula for the quotient rule?
If h(x) = u(x)/v(x), then h’(x) = (vu’-uv’)/v^2.
What does the term ‘one-to-one function’ mean?
A function is one-to-one if it never takes the same value twice.
What is the derivative of f(x) = cos(x^2)?
-2xsin(x^2)
What is the relationship between the graphs of a function and its inverse concerning their slopes?
The slopes are reciprocals of each other at corresponding points.
Steps to find the inverse of f(x)
- Replace f(x) with y .
- Swap x with y
- Rearrange the function algebracally equal to y
- Finally, replace y with f^−1(x)
Definiton of a Derivative at a point “a”
f’(a) = lim h->0 (f(a + h) - f(a)) / h
Definition of a derivative as a function
f’(x) = lim h->0 (f(x - h) - f(x)) / h
Whats the reciprocal?
1 divided by that number
d/dx sin x =
cos x
d/dx cos x =
-sin x
d/dx tan x =
sec^2 x
d/dc csc x =
(-csc x)(cot x)
d/dx sec x =
(sec x)(tan x)
d/dx cot x =
-csc^2 x
d/dx a^x =
a^x ln a
d/dx loga x =
1 / x ln a
logarithmic differentiation
Sometimes, it can be helpful to take the log of both sides then use implicit differentiation to solve
Solve y = (x^3 - x + 1)^4 using the chain rule
y = 4(x^3 - x +1)(3x^2 - 1)
Derivative of a constant multiple
d/dx cu = c * d/dx u
units of the derivative of particle motion are…
units of y / units of x
If s(t) is the position function of an object, then s’(t) gives:
v(t) gives the object’s velocity
If s(t) is the position function of an object, then s’‘(t) gives:
a(t) gives the object’s accelaration
If s(t) is the position function of an object, then s’’‘(t) gives:
j(t) gives the object’s jerk
Displacement
Overall change in position
How much ground the object covered is…
total distance
What does velocity tells you?
how fast you’re going and in what direction
What does speed tells you?
How fast you’re going without regard to direction
How to find speed?
Take the absolute value of velocity (v(t))
Positive velocity is:
A particle moving forward/right on a horizontal line
Negative velocity is:
A particle moving backwards/left on a horizontal line
v(t) = 0 means
Particle is at rest
Formula for avg. v(t) on [a,b]
(s(b) - s(a)) / b - a
A particle is speeding up when…
velocity and acceleration have the same sign
A particle is slowing down when…
velocity and acceleration have different signs
Steps to create a sign chart for particle motion
- Find Zeros, set function = 0
- Mark Zeros on a Number Line
- Substitute each test interval bwt zeros into the function
- Determine if its + or - in that interval
What info does a sign chart gives you about the motion of a particle?
helps analyze changes in direction and speed based on where there are zeros
Formula for tangent line at x = a
y = f(a) + f’(a)(x - a)
formula for the normal line at x = a
y = f(a) - (1/f’(a))(x - a)
d/dx e^x =
e^x
d/dx arcsin x =
1 / √(1-x^2)
d/dx arccos x =
-1 / √(1-x^2)
d/dx arctan x =
1 / (x^2 + 1)
d/dx arccot x =
-1 / (x^2 + 1)
d/dx arcsec x =
1 / (|x|•√(x^2 - 1))
d/dx arccsc =
-1 / (|x|•√(x^2 - 1))
