Differentiation Flashcards
Simple diff
Differentiate a x^n
an x^(n-1)
Chain rule
y = a(exp)^n
dy/dx = na(exp)^(n-1) . d/dx (exp)
Product rule
y=uv
dy/dx = v du/dx + u dv/dx
Quotient rule
y=u/v
v du/dx - u dv/dx dy/dx = -—————————
v^2
Grad of tang and normal
1
Grad of normal = - ————
dy/dx
Eqn of a line with the grad
y - y1 = m (x - x1)
2 tests to find nature of stationary pt
x (20/3)- (20/3) (20/3)+
(3x-20) -ve 0 +ve
(x+10)
dv/dx =
-π(3x-20) +ve. 0. -ve
(x+10)
tangent / —— \
Conclusion Maximum pt
OR
d^2y
——— > 0 ——— min pt
dx^2
d^2y
——— < 0 ——— max pt
dx^2
d
—— sinx =
dx
d
—— cosx =
dx
d
—— tanx =
dx
d
—— tan^n f(x) =
dx
d
—— sinx = Cosx
dx
d
—— cosx = -sinx
dx
d
—— tanx = Sec^2 (x)
dx
d
—— tan^n f(x) = n tan^(n-1) f(x) X d/dx [tan f(x)]
dx
^ same for others trigo func
d
—— e^x =
dx
d
—— e^f(x) =
dx
d
—— e^x = e^x
dx
d
—— e^f(x) = e^f(x) d/dx [f(x)]
dx
d
—— ln f(x) =
dx
d d/dx [f(x)]
—— ln f(x) = —————
dx [f(x)]
log (x) = convert to ln
b
y
x = convert to e
lnx
log (x) = ——
b lnb
y y ln ( x ) x = e
Differentiate
Cotx
Secx IN Mf26
Cosecx IN Mf26
And what to do when it’s eg cot [f(x)]
Differentiate
Cotx = -cosec^2 x
Secx = Sec x Tan x
Cosecx = - cosec x cot x
cot [f(x)] = [-cosec^2 f(x)] f’(x)
Differentiate
sin^-1 [f(x)]
Cos^-1 [f(x)]
Tan^-1 [f(x)]
And ‘range’ f(x) must be in for ^ all 3
Differentiate
f’(x)
sin^-1 [f(x)] = ————————
√1 - [f(x)]^2
(Sq root is for entire denominator)
f’(x) Cos^-1 [f(x)] = - ————————
√1 - [f(x)]^2
(Sq root is for entire denominator)
f’(x) Tan^-1 [f(x)] = ————————
1 + [f(x)]^2
Implicit diff - finding eqn from 2 implicit eqns
If u see any trig exp having 1/2Θ, use __
Cos2A and Sin2A exp in MF26
