df(x0)/dx into lim formula
lim((f(x0 + change in x) - f(x))/change in x) where change in x -> 0
compute the derivative of x^3 at x=a
3a^2
dC/dx
0, where C is constant
dx^p/dx
px^(p-1)
where p != 0 and is a real number
d/dx(sinx)
cosx
d/dx(cosx)
-sinx
de^x/dx
e^x
d/dx(ln(x))
1/x
4 derivative rules
- linearity
- chain rule
- product rule
- quotient rule
linearity formula
- d/dx[f(x) + g(x)] = df(x)/dx + dg(x)/dx
- d/dx[cf(x)] = c df(x)/dx
chain rule
if z = f(y) and y = g(x) then
dz/dx = d/dx[f(g(x))] = df(y)/dy . dg(x)/dx
product rule
u’v + uv’
quotient rule
u’v - uv’/v^2
what does differentiable imply
if a function is y=f(x) is discontinuous at x=x0, then it is not differentiable
what are higher order derivatives
takes multiple differntiations
differentiate distance
distance -> speed -> acceleration
differentiate displacement
displacement -> velocity -> acceleration
l’hoptial’s rule
limx->a(f(x)/g(x)) = limx->a(f’(x)/g’(x))
