Technik des Differenzierens

Faktorregel (d/dx)[c f(x)] = c[(d/dx)f(x)]

Summenregel (d/dx)(u + v) = (du/dx + dv/dx)

Produktregel (d/dx)(u v) = (du/dx)*v + (dv/dx)*u (u v)' = u'v + v'u

Potenzregel dx^n/dx = n * x^(n-1); n positiv, ganz

Quotientenregel (d/dx)(u/v) = (1/v^2)(v*(du/dx)-u*(dv/dx)) (u/v)' = (u'v-uv')/v^2

Kettenregel dy/dx = (dy/dt)*(dt/dx) oder f'(x)=f'(t)fi'(x)

Logarithmische Ableitung (d/dx)f(x) = f(x)*(d/dx)*ln(f(x))

Umkehrregel fi'(y) = 1/f'(x) dx/dy = 1/(dy/dx)

Ableitung einer Funktion in Parameterdarstellung dy/dx = (dy/dt)/(dx/dt) oder f'(x) = psi'(t)/fi'(t)

Ableitung spezieller Funktionen

de^x/dx=e^x

(d/dx)a^x = a^x * ln(a)

(d/dx)log a(x) = 1/(x * ln(a)) = (1/x) * log a(e)

(d/dx)ln(x) = 1/x

(d/dx)sin(x) = cos(x)

(d/dx)cos(x) = -sin(x)

(d/dx)tan(x) = 1/cos^2(x) = 1 + tan^2(x)

(d/dx)cot(x) = - 1/sin^2(x) = -(1+cot^2(x))

(d/dx)arcsin(x) = 1/(sqrt(1-x^2))

(d/dx)arccos(x) = -1/(sqrt(1-x^2))

(d/dx)arctan(x) = 1/(1+x^2)

(d/dx)arccot(x) = -1/(1+x^2)

(d/dx)sinh(x) = cosh(x)

(d/dx)cosh(x) = sinh(x)

(d/dx)tanh(x) = 1/(cosh^2(x))

(d/dx)coth(x) = -1/(sinh^2(x))

(d/dx)arsinh(x) = 1/(+sqrt(x^2 + 1))

(d/dx)arcosh(x) = 1/(+-sqrt(x^2-1)); x > 1

(d/dx)artanh(x) = 1/(1-x^2); |x| < 1

(d/dx)arcoth(x) = 1/(1-x^2); |x| > 1