# Given the expression e^x + x^3 - x^2 + x prove that the derivative of te expression is positive.

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Derivative of (e^x + x^3 - x^2 + x) is

(e^x + x^3 - x^2 + x)' = e^x+(3x^2-2x+1) .

= e^x+2x^2 +(x-1)^2. Obviously each of these 3 terms are positive.

Hence the derivative of (e^x+x^3-x^2+x) is always posiive.

We'll associate a function to the given expression:

f(x) = e^x + x^3 - x^2 + x

Let's calculate

f'(x)=(e^x + x^3 - x^2 + x)'

f'(x)=e^x+3x^2-2x+1

f'(x)=e^x+2x^2+x^2-2x+1

We can combine the last 3 terms:

f'(x)=e^x+2x^2+(x-1)^2>0

e^x>0 (1)

2x^2>0 (2)

(x-1)^2>0 (3)

If we'll add (1),(2),(3):

e^x+2x^2+(x-1)^2>0

So, f'(x) it's obviously positive.

The derivative of the expression e^x+x^3-x^2+x is e^x+3x^2-2x+1

Now e^x+3x^2-2x+1= e^x+2x^2+x^2-2x+1= e^x+2x^2+(x-1)^2

e^x is positive whether x is positive of negative. x^2 also is always positive irrespective of whether x is positive or negative. (x-1)^2 is also never negative irrespective of x being negative or positive.

Therefore the expression is the sum of three positive terms and therefore positive.