Differentiation Flashcards
Limit Definition of the Derivative
If f’(x) = lim(h->0) [(f(x+h)-f(x))/h]
Combinations of Functions
(cf)’
(cf)’ = c*f’
Combinations of Functions
(f+g)’
f’+g’
The Product Rule
(uv)’ = u’v+u*v’
The Quotient Rule
(u/v)’ = (vu’-v’u)/v²
The Chain Rule
(f(u(x))’ = u’(x)*f’(u(x))
Extended Product Rule
(u1 * u2 * u3 * … * un)’
= u1’u2u3…un + u1u2’u3…un + u1u2u3’…un + …
Leibniz’s Rule
(fg)’ = f’g + fg’
(fg)’’ = f’‘g + f’g’ + f’g’ + fg’’
(f*g)’’’ = f’'’g + f’‘g’ + f’g’’ + fg’’’
etc.
Differentiate
f(x) = x^n
f’(x) nx^(n-1)
Differentiate
f(x) = x^(a/b)
f’(x) = (a/b)*x^(a/b -1)
Mean Value Theorem
-If f is continuous on some interval [a,b] and differentiable on (a,b), then there is a number c in (a,b) such that:
f’(c) = (f(b) - f(a))/(b-a)
-c is the point on f(x) where f’(x) is equal to the gradient of a straight line drawn between a and b
Maximum and Minimum Points
If c is a maximum or minimum point for a function f and if f is differentiable at c, then
f’(c) = 0
Rolle’s Theorem
Suppose f(a) = f(b), then if f is continuous on [a,b] and differentiable on (a,b) there exists a c in (a,b) such that f’(c) = 0
Couchy’s Mean Value Theorem
If f & g are continuous on [a,b] and differentiable on (a,b) such that:
f’(c) / g’(c)
= (f(b) - f(a)) / (g(b) - g(a))
L’Hopital’s Rule
if f(a) = g(a) = 0, then:
lim(x->a) f(x)/g(x)
= lim(x->a) f’(x)//g’(x)
