DIFFERENTIAL CALCULUS

Question 1

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.

Answer 1

DIFFERNTIAL CALCULUS

Question 2

TRIGONOMETRIC DERIVATIVES
sin 
cos
tan
cot
csc
sec

Answer 2

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

Question 3

derivative of inverse TRIGONOMETRIC function

Answer 3

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)]

Question 4

derivative of hyperbolic fnc.

Answer 4

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

Question 5

derivative of inverse hyperbolic fnc.

Answer 5

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)]

Question 6

d/dx [Logb (x)] =

d/dx [ln (x)] =

Answer 6

d/dx [Logb (x)] = 1/ x ln (b)

d/dx [ln (x)] = 1 / x

Question 7

The ______ is if the two “parts” of the function are being multiplied together, and the chain rule is if they are being composed.

Answer 7

PRODUCT RULE

Question 8

the ___ is a method of finding the derivative of a function that is the ratio of two differentiable functions.

Answer 8

QUOTIENT RULE

Question 9

QUOTIENT RULE .

Answer 9

d/dx (u/v) = [ v(du/dx) - u (dv/dx) ] / v^2

Question 10

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.

Answer 10

chain rules

dy/dx = dy/du du/dx

Question 11

In mathematics, a _____ of a function of several variables is its derivative with respect to one of those variables, with the others held constant.

Answer 11

partial derivative

Question 12

differentiating both sides of the equation with respect to x and then solving the resultsing equation of y

Answer 12

implicit diferentionation

Question 13

how to solve/derive an implicit eq.

Answer 13

solve both side in terms of x, the derivative of y variable would be dy/dx.

equate dy/dx with the other terms.

Question 14

difference of derivative and integration

Answer 14

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.

Question 15

An is an expression involving two functions whose limit cannot be determined solely from the limits of the individual functions.

Answer 15

indeterminate form

Question 16

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.

Answer 16

L’Hôpital’s rule or L’Hospital’s rule

0/0 or ∞/∞

Question 17

L’Hôpital’s rule or L’Hospital’s rule equation

Answer 17

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

Question 18

if the indeterminate form is in product form, can you use lhopital’s rule to solve?? like 0∞ ?

Answer 18

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…

Question 19

if the indeterminate form is in difference form, can you use lhopital’s rule to solve?? like ∞-∞ ?

Answer 19

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)]

Question 20

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 ∞).

Answer 20

indeterminate power

Question 21

how to convert power form of indeterminate form to a fraction form (0/0 or ∞/∞ for us to apply Lhopital’s rule.)

Answer 21

since y= f(x)^g(x)

use; y= g(x) ln f(x)

Question 22

how to get the slope of a tangen line. (m) and tangent equation

Answer 22

take the first derivative

m=y’

to get the eq. just subtitue
(y-yt)= m (x-xt)

Question 23

a line through teh tangency and perpedicular to the tangent line

Answer 23

normal lline, kind of making an X with the tangent line

Question 24

how to get the slop of a normal line??

Answer 24

m???

m= -1/y’
perpendicular to the tangent line`

Question 25

____________is a point in the domain of the function where the function is either not differentiable or the derivative is equal to zero.

Answer 25

a critical point

Question 26

getting the first derivative of a function is equal to the?

Answer 26

slope

Question 27

getting the 2nd derivative of a function is tell us what about the slope?

Answer 27

if the slop is increasing or decreasing.

Question 28

how to tell if the slop is positive or negative in terms of 2nd derivative

Answer 28

increases - positive -

decreases - negative-

Question 29

a point on a smooth plane curve at which the curvature changes sign.

Answer 29

inflection point

Question 30

getting an indeterminate after getting the first derivative of a function means it is the ?

Answer 30

inflection point

Question 31

getting the 2nd derivative of the inflection point will give us the

Answer 31

zero