How to find the first and second derivative?

Question

find the first and second derivative of; f(x) = e^((x^2) - (x^4))

any help will be much appreciated :)

Answer 1

the derivative of e^u is e^u du

f(x) = e^(x^2 - x^4)

f '(x) = e^(x^2 - x^4) (2x - 4x^3)

rearrange...

f '(x) = (2x - 4x^3) e^(x^2 - x^4)

that's the first derivative. For the second you'll have to use the product rule...

if y = uv, then y' = uv' + u'v

so...

(2x - 4x^3)(e^(x^2 - x^4))(2x - 4x^3) + (e^(x^2 - x^4))(2 - 12x^2)

simplify. take out a e^(x^2 - x^4) term

e^(x^2 - x^4) [ (2x - 4x^3)^2 + (2 - 12x^2) ]

simplify what's in the brackets and move out front...

(16x^6 - 16x^4 - 8x^2 + 2) e^(x^2 - x^4)

messy, I know. but it's really not that bad

Answer 2

f(x) = e^((x^2) - (x^4))

f'(x)=e^((x^2) - (x^4)) (2x-4x^3)

Use the product rule to find f''(x)

f''(x)=e^((x^2(-(x^4)) (2-12x^2) + (2x-4x^3) e^((x^2)-(x^4)) (2x-4x^3)

f''(x)= e^((x^2(-(x^4)) (2-12x^2) + (2x-4x^3)^2 e^((x^2)-(x^4))