Find out the number of real roots of equation $f'(x) = 0$, where    $$f(x)=(x-1)(x-2)(x-3)(x-4)(x-5)$$    How can I differentiate this function without expanding it to the polynomial form. Am I underestimating some theory of equation concept associated with it? (I know the product rule approach and solving by simplifying but I want to know is there any other way to solve it)
Let   $$f(x) = (x-1)(x-2)(x-3)(x-4)(x-5),\quad -\infty&lt; x&lt;\infty.$$   The number of distinct roots of equation   $$\frac{d}{dx}f(x) = 0$$   is exactly ?     The only method that I know is to multiply and then find the derivative of function then apply Sturm's theorem but it seems vague when you have to solve the question in 3 to 5 minutes . So you are requested to suggest a plausible alternative approach.
Double integral with transformation; possible error in limits I'm asked to show that the following integral is true with the transformation $u = x+y$ and $y=uv$:  $$\int_0^1 \int_0^{1-x} e^{y/(x+y)} dx dy = \frac{e-1}2 $$  I found the determinant of the jacobian of the transformation is $-u$, and the exponent is just $v$ so the integral becomes $$\iint_R |-u|e^v du dv = \frac {u^2}2 e^v|_R$$ $$=\frac{(x+y)^2}2e^{y/(x+y)}|_{x=0}^{1-x}\;|_{y=0}^1$$ And plugging it all in doesn't work (AFAICT). For one, the x's don't all cancel.  I've looked at it 100 times. It's definitely written dxdy not dydx even though there is definitely an x in the integral. Maybe this is a typo and why my calculation isn't working out? I tried reversing the order to dydx and it didn't work. I tried changing the limit of integration to $1-y$ and it didn't work out.  I'm not even sure how to figure out the domain of integration in the uv-plane. Flipping things around yields y=uv and x=u-uv. 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