produktregel-beispiel

$$||"id":737579265041||f(x)=(3x+x^2)\ast (x-1)(3x+x^2)=u(x)(x-1)=v(x)u^{\prime }(x)=3+2xv\ast (x)=1f^{\prime }(x)=u^{\prime }(x)\ast v(x)+v^{\prime }(x)\ast u(x)f^{\prime }(x)=3x+2x^2-3-2x+3x+x^2-4x+3x^2-3=3x^2+4x-3$$ $$||"id":721248608012||g(x)=(3x+x^2)\ast \sin (x)u^{\prime }(x)=3+2x{v}^{\prime }(x)=\cos (x)g^{\prime }(x)=(3+2x)\ast \sin (x)+\cos (x)\ast (3x+x^2)$$

3.47) a) -c) a)

$$||"id":1028083841554||f(x)=(2+x)\ast (1+x^2)u^{\prime }(x)=2x+2v^{\prime }(x)=1+2x{f}^{\prime }(x)=(1)\ast (1+x^2)+(2x)\ast (2+x)=1+x^2+4x+2x^2=3x^2+4x+1$$

b)

$$||"id":55428408514||f(x)=(x-1{)}^2\ast \sqrt{x}=(x^2-2x+1)\ast x^{\frac{1}{2}}{f}^{\prime }(x)=$$

c)

$$||"id":608757675152||f(x)=x\ast e^x{f}^{\prime }(x)=1\ast e^x+e^x\ast x=e^x\ast (1+x)$$