Continuity and Differentiability — Formula Sheet

lim_{x \to c} f(x) = f(c)

Continuity at a = lim_{x \to a^+} f(x) = f(a)

Continuity at b = lim_{x \to b^-} f(x) = f(b)

f'(c)=lim_{h \to 0} [f(c+h)-f(c)]/h

(u \pm v)'=u' \pm v'

(uv)'=u'v+uv'

(u/v)'=(u'v-uv')/v^2

d/dx(x^n)=nx^{n-1}

d/dx(sin x)=cos x

d/dx(cos x)=-sin x

d/dx(tan x)=sec^2 x

d/dx(e^x)=e^x

df/dx=dv/dt · dt/dx

df/dx=d(w∘u)/dt · dt/dx = dw/ds · ds/dt · dt/dx

dy/dx=(dy/dt)/(dx/dt)

dy/dx=g'(t)/f'(t)

If f = v∘u and t = u(x), then df/dx = (dv/dt)(dt/dx).