Question

whats the difference between prime, derivative, antiderivative, integration, i ask since its costing me major points on test when i mix them up (i.e. with trig-when using sin and cos)

Answer 1

prime functions, antiderivatives, and indefinite integrals are all the same -- they do the opposite of differentiation

Answer 2

prime is derivative...

single prime = first derivative
double prime = second derivative
and so on

Answer 3

antiderivative is integration (indefinite)

Answer 4

definite integrals however map functions to numbers; they give what is essentially *signed* area bounded by a function:$$\int_a^b f(x)\,dx$$their relationship with the antiderivative \(\int f(x)\,dx\) is the fundamental theorem of calculus, which says that for \(F=\int f(x)\,dx\) and \(a\le b\) we have:$$\int_a^b f(x)\,dx=F(b)-F(a)$$

Answer 5

so i can safely say that y prime of sin is cos which is -sin which is -cos which is sin
this is also true if I'm deriving and integrating
but if I'm finding the antiderivative the its backwards? as in sin to -cos to -sin to cos
yes?

Answer 6

if y = sin(x) then y' = cos(x)
the antiderivative of cos(x) is sin(x) +C (a constant to account for initial conditions).

You lost me on the rest... I don't understand what you're saying

Answer 7

please never say 'y prime of ...' that makes no sense... the derivative of \(\sin x\) with respect to \(x\) is \(\cos x\)

Answer 8

if \(y=\sin x\) then you can say the derivative of \(y\) i.e. \(y'\) is \(\cos x\), i.e. \(y'=\cos x\)

Answer 9

the reason for the constant of integration is that the derivative only gives you information about rates of change (i.e. relative values), not absolute values, so for example the following two functions have the same derivative despite being in different parts of space: