Differentiation Rules (Chapter 3) Flashcards
What is the d/dx of ex?
ex
What is the d/dx of cos x?
-sin x
How do you find the derivative of f at x?
Lim as delta x -> 0: f(x + delta x) - f(x) / delta x
What is the d/dx of eu?
euu’
What is the d/dx of ln x?
1 / x
What is the d/dx of logax?
1 / (ln a) * x
What is the d/dx of cos-1(u)?
-u’ / sqrt(1-u2)
What is the d/dx of sec-1(u)?
u’ / |u| * sqrt (u2-1)
What is the d/dx of |u|?
u / |u| * (u’)
What is the d/dx of tan x?
sec2x
What is the d/dx of ax?
(ln a) * ax
What is the formula for volume of a conic?
1/3(pi)r2h
What is the quotient rule ([d/dx] u / v)?
vu’ - uv’ / v2
What is the alternate definition of a limit? How can this be used to determine differentiability with absolute value?
f(x) - f(x) / x - c
Evaluate the limit at c+ and c-. If they differ, the function is not differentiable.
What is the d/dx of ax?
(ln a) ax
What is the chain rule ([d/dx] f(u))?
f’(u) * u’ for each function in the expression.
What is the d/dx of csc-1(u)?
-u’ / |u| sqrt(u2 - 1)
What is the d/dx of au?
(ln a) au * u’
What is the d/dx of sin x?
cos x
What is the product rule ([d/dx] u * v)?
uv’ + vu’
What is the simple power rule ([d/dx] xn)?
For xn, nxn-1.
For x1, 1.
What is the derivative of a constant?
0
What is the d/dx of sec x?
sec x tan x
What is the constant multiple rule ([d/dx] c*u)?
c * u’
What is the d/dx of csc x?
-csc x cot x
What is the d/dx of sin-1(u)?
u’ / sqrt(1 - u2)
What is the d/dx of cot-1(u)?
-u’ / 1 + u2
What is the d/dx of cot x?
-csc2x
What is the d/dx of logax?
1 / (ln a) x
What is the sum / difference rule ([d/dx] u + v / u - v?)
u’ + v’ / u’ - v’
What is the d/dx of tan-1(u)?
u’ / 1 + u2
What is the General Power Rule ([d/dx] un)?
nun-1 * u’
What is the d/dx of ln u?
u’ / u
What is the d/dx of logau?
u’ / (ln a) u
What would you use to find the d/dx of (x-2)^2 / sqrt(x^2 + 1)?
Logarithmic differentiation. Take Ln of both sides. Use ln u for ln y. Use Ln division rule Ln (x/y) Ln x - Ln y.
How do you find the derivative of an inverse function?
It is the reciprocal of f’(g(x)) where g(x) is the inverse function.
A conical tank with vertex down is 20 feet across the top and 16 feet deep. How can we find:
a. Water is flowing into the tank at a rate of 5 cubic feet per minute. How can we find the rate of change of the depth of the water when the water is 4 feet deep?
b. If the water is rising at a rate of 2/3 feet per minute when h = 8, how can we determine the rate at which water is being pumped into the tank?
Given 1/3 pi r^2 h:
a. Using similar triangles, find r = some value of h and back subtitute into the above formula (making sure to square and combine substituted h into h already present).
r/h = 10/16, 16r = 10h, r=10/16h = 5/8h
1/3 pi (5/8h)^2 h = 1/3 pi 25/64 h^3
Then, find derivative of modified formula, being sure to use implicit differentiation for h. This is dv/dt.
dv/dt = 75/192 * pi * h^2 * h’ = 25/64 * pi * h^2 * h’
Equate dv/dt with modified formula, plug in h (4) and solve for h’.
5 = 25/64 * pi * 4^2 * h’; h’ = 4/5pi ft/min
A conical tank with vertex down is 20 feet across the top and 16 feet deep. How can we find:
b. If the water is rising at a rate of 2/3 feet per minute when h = 8, how can we determine the rate at which water is being pumped into the tank?
Using the formula already determined from item a, (dv/dt = 75/192 * pi * h^2 * dh/dt), plug in dh/dt (2/3) and h.
What is Newton’s method?
- Graph function.
- Approximate Zeros visually by guessing a value of x close to the zero.
- Draw a table with 5 columns:
n | xn | f(xn) | f‘(xn) | xn - f(xn) / f’(xn)
- Compute all values. For row 1, xn is your guessed value. For subsequent rows, xn is the value from column 5 in the prior row.
