Unit 1 Flashcards
What does sin x differentiate to?
Cos x
What does cos x differentiate to?
-Sin x
What does tan x differentiate to?
2
sec x
What does cosec x differentiate to?
- cosec x Cot x
What does sec x differentiate to?
Sec x Tan x
What does cot x differentiate to?
2
-cosec x
why is sinx/cosx equal to?
tan x
what is 1/sin x equal to
cosec x
what is 1/cos x equal to
sec x
what is 1/tan x equal to
cos x/ sin x = cot x
-1
What does sin x differentiate to?
2
1/✔️1-x
✔️ = square root
- 1
What does cos x differentiate to?
2
-1/✔️1-x
✔️ = square root
-1
What is the does tan x differentiate to?
2
1/1+x
What is the product rule
Y = UV
dy/dx = U’V + UV’
What is the quotient rule
y = u/v
2
dy/dx = U’V-UV’/V
What does. 2. differentiate to
x
e
2
x
2x e
how do you differentiate exponential
you times the exponential by the derivative of the power while the power remains constant
Ex/
3
x
e
becomes
3 2. x 3x e
how do you differentiate natural logs
ln x
1 is divided by the x part and then times by the differentiation of it
Ex/ ln 3x - 1
=. 1/3x - 1 times by 3
=. 3/3x - 1
how do you differentiate implicitly
you differentiate the y normally (as if it was an x) then you add a dy/dx after it
Ex/. 2
d/dx y. = 2y dy/dx
3. 2
d/dx. y. = 3y. dy/dx
d/dx. sin x = cos x dy/dx
y. y d/dx e. =. e dy/dx
d/dx ln y =. 1/y. dy/dx
how do you implicitly differentiate the equation of a tangent
differentiate implicitly then rearrange so you have a dy/dx = equations
then sub in the values for x and y ( given by the coordinates in the question)
this gives the gradient of the tangent (m)
sub these values into y-b = mx-a ( with the y coordinated being b and the x coordinate being a) to get the equations of the tangent
what are the log rules
ln ab = ln a + ln b
ln a/b = ln a - ln b
n ln a = n ln a
how do you logarithmicly differentiate
take a natural log of both sides, you then simplify through the log rules and differentiation
what is the formula for parametric differentiation
dy/dt
dy/dx = ———
dx/dt
how do you parametrically differentiate
differentiate the x term
differentiate the y term
you then put them together through the parametric differentiation equation [ (dy/dt)/(dx/dt) ]
this gives you your final equation
how’s do you integrate exponentials
integrated normally but decide by the derivative of the power of the exponential
Ex
3x+2 f. e. dx
3x+2 =. 1/3e. + c
how do you standard integrate with the chain rule
add one to the power of equation, divide by that then divide by the derivative of the equation
Ex/
4
f (3x +2 ). dx
5
= 1/5 X 1/3 X (3x +2 ) + c
5 = 1/15. (3x +2 ). + c
How do you integrate logs
You change the 1/x back into the ln | x | ( making sure to have vertical lines in the ln ) then divide by the derivative of the x
Ex/
f 1/9x + 5 dx
= 1/9 ln | 9x + 5 | + c
what is the formula for integration by parts
f U V’ = U V - f U’ V
what is an example of integration by parts
I = f x sin3x dx
u = x. v’ = sin3x u’ = 1 v = -1/3 cos3x
I = uv - f u’v dx I = -1/3 x cos3x - f -1/3 cos3x dx I = -1/3 x cos3x + f 1/3 cos3x dx I = -1/3 x cos 3x + 1/3 X 1/3 X sin 3x + c I = -1/3 x cos 3x + 1/9 sin 3x + c
