Differentiation Flashcards
What is differentiation from first principles?
(Expression)
F’(x) = lim f (x+h) - f (x)
———————
h
What do different values of derivatives tell us?
F’(x) < 0: decreasing
F’(x) = 0: stationary point
F’(x) > 0: increasing
What do different values of second derivatives tell us?
F”(x) < 0: concave/max
F”(x) = 0: point of inflection
F”(x) > 0: convex/min
Derivative of ax^n
anx^(n-1)
Derivative of a^x
ln a • a^x
Derivative of e^x
e^x
Derivative of ln x
1/x
Derivative of Sin kx
K Cos kx
Derivative of Cos kx
-k Sin kx
Derivative of Tan x
Sec^2 x
Derivative of Sec x
Sec x Tan x
Derivative of Cot x
—Cosec^2 x
Derivative of Cosec x
Cosec x Cot x
What is the chain rule?
f (x) ~> f ‘(x) • x’
What is quotient rule?
u/v ~> vu’-uv’
———
v^2
What is 1/(dx/dy) ?
Dy/dx
How to find dy/dx when given dy/dt and dx/dt?
Divide dy/dt by dx/dt, this will give
dy/dt X dt/dx so the dts cancel leaving dy/dx
What is product rule?
uv ~> uv’ + u’v
What is the implicit differentiation rule?
F(y) ~> f ‘(y) X dy/dx
What is rate of change?
How do you connect them?
Rate means over dt
dv = dv X …
dt … dt
