DIFFERENTIAL CALCULUS Flashcards
is a subfield of calculus that studies the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus—the study of the area beneath a curve.
DIFFERNTIAL CALCULUS
TRIGONOMETRIC DERIVATIVES sin cos tan cot csc sec
d/dx of the ff: sin x = cos x cos x = -sinc x tan x = sec^2 x cot x = -csc^2 x csc x = -csc x cot x sec x = sec x tan x
derivative of inverse TRIGONOMETRIC function
d/dx of the ff:
sin^-1 x : 1 / √(1-x^2)
cos^-1 x: -1 / √(1-x^2)
tan^-1 x: 1 / (1+x^2)
cot^-1 x: - 1 / (1+x^2)
csc^-1 x: -1 / [x√(x^2-1)]
sec^-1 x: 1 / [x√(x^2-1)]
derivative of hyperbolic fnc.
d/dx of the ff:
sinh x = cosh x
cosh x = sinh x
tanh x = sech^2 x
coth x = -csch^2 x
csch x = -csch x coth x
sech x = -sech x tanh x
derivative of inverse hyperbolic fnc.
d/dx of the ff:
sinh^-1 x : 1 / √(1+x^2)
cosh^-1 x: 1 / √(x^2-1)
tanh^-1 x: 1 / (1-x^2)
coth^-1 x: 1 / (1-x^2)
csch^-1 x: -1 / [|x|√(x^2+1)]
sech^-1 x: -1 / [x√(1-x^2)]
d/dx [Logb (x)] =
d/dx [ln (x)] =
d/dx [Logb (x)] = 1/ x ln (b)
d/dx [ln (x)] = 1 / x
The ______ is if the two “parts” of the function are being multiplied together, and the chain rule is if they are being composed.
PRODUCT RULE
the ___ is a method of finding the derivative of a function that is the ratio of two differentiable functions.
QUOTIENT RULE
QUOTIENT RULE .
d/dx (u/v) = [ v(du/dx) - u (dv/dx) ] / v^2
In calculus, the _____ is a formula that expresses the derivative of the composition of two differentiable functions f and g in terms of the derivatives f and g.
chain rules
dy/dx = dy/du du/dx
In mathematics, a _____ of a function of several variables is its derivative with respect to one of those variables, with the others held constant.
partial derivative
differentiating both sides of the equation with respect to x and then solving the resultsing equation of y
implicit diferentionation
how to solve/derive an implicit eq.
solve both side in terms of x, the derivative of y variable would be dy/dx.
equate dy/dx with the other terms.
difference of derivative and integration
Differentiation is used to calculate the gradient of a curve. It is used to find out the instant rates of change from one point to another. Integration is used to calculate the area under or between the curves.
derivative = rate of change.
integrate- amount of change.
An is an expression involving two functions whose limit cannot be determined solely from the limits of the individual functions.
indeterminate form
In mathematics, more specifically calculus, ___ is a theorem which provides a technique to evaluate limits of indeterminate forms. what specific indeterminate forms are required to apply l hopital’s rule.
L’Hôpital’s rule or L’Hospital’s rule
0/0 or ∞/∞
L’Hôpital’s rule or L’Hospital’s rule equation
if the initial solution is in indeterminate form then
lim(x→a) f(x)/g(x) = lim(x→a) f’(x)/g’(x)
*get the limit of the derived function of each
if the indeterminate form is in product form, can you use lhopital’s rule to solve?? like 0∞ ?
yes, arrange the equation such that it would be in fraction form, then you can apply lhoptial;s rule
f(x)g(x) = f(x) / 1/g(x) like these…
if the indeterminate form is in difference form, can you use lhopital’s rule to solve?? like ∞-∞ ?
yes just rearrange to fraction of two fnc.
use;
f(x) - g (x) = [f^2(x)g(x) - g^2(x)f(x)] / [f(x)g(x)]
is what we call the form of a limit lim(x→a) f(x)^g(x)
where the limit has the form 0^0 or 1^∞ or ∞^ 0 when you plug in the value a (which might be ∞).
indeterminate power
how to convert power form of indeterminate form to a fraction form (0/0 or ∞/∞ for us to apply Lhopital’s rule.)
since y= f(x)^g(x)
use; y= g(x) ln f(x)
how to get the slope of a tangen line. (m) and tangent equation
take the first derivative
m=y’
to get the eq. just subtitue
(y-yt)= m (x-xt)
a line through teh tangency and perpedicular to the tangent line
normal lline, kind of making an X with the tangent line
how to get the slop of a normal line??
m???
m= -1/y’
perpendicular to the tangent line`
____________is a point in the domain of the function where the function is either not differentiable or the derivative is equal to zero.
a critical point
getting the first derivative of a function is equal to the?
slope
getting the 2nd derivative of a function is tell us what about the slope?
if the slop is increasing or decreasing.
how to tell if the slop is positive or negative in terms of 2nd derivative
increases - positive -
decreases - negative-
a point on a smooth plane curve at which the curvature changes sign.
inflection point
getting an indeterminate after getting the first derivative of a function means it is the ?
inflection point
getting the 2nd derivative of the inflection point will give us the
zero
