Product rule and a zero gradient

Practise product rule and a zero gradient with an original VectorPaths question, worked explanation and related VCE Methods practice.

3 practice marks

Original practice question

Let $f(x)=(x^2+1)e^x$. (a) Differentiate and factorise the result. [2 marks] (b) Find every x at which the gradient is zero. [1 marks]

Worked explanation

• Part (a)

  The product rule gives $2xe^x+(x^2+1)e^x=e^x(x^2+2x+1)=e^x(x+1)^2$.

• Part (b)

  $e^x>0$ for real x, so the gradient is zero exactly when $(x+1)^2=0$.

(a) $f\prime(x)=e^x(x+1)^2$.; (b) $x=-1$.

Original VectorPaths practice; not an official VCAA question or marking rubric.