what is the 3rd derivative of f(x)=x^2 e^x^2
let me see what you got for the first derivative, so I can see if you are on the right track.
2x e^x^2+x^2 ex^2
no, you have a product rule
y = (x^2)(e^(x^2))
y' = (x^2)(2x)(e^(x^2)) + (e^(x^2)) (2x)
now factor this
y' = (2x)(e^(x^2)) (x^2 + 1)
now you have triple product, so
y'' = (2x)(e^(x^2)) (2x) + (2x)(x^2 + 1)(2x(e^(x^2)) + 2(e^(x^2)) (x^2 + 1)
= 2 e^(x^2) (2x^2 + 2x^2(x^2+1) + (x^2 + 1) )
= 2 e^(x^2) (5x^2 + 2x^4 + 1)
now carefully do it once more
To find the third derivative of the function f(x) = x^2 * e^(x^2), we can use the chain rule and the product rule.
Step 1: Compute the first derivative of f(x):
f'(x) = (2x * e^(x^2)) + (x^2 * (2x * e^(x^2)))
= 2x * e^(x^2) + 2x^3 * e^(x^2)
Step 2: Compute the second derivative of f(x):
f''(x) = 2 * e^(x^2) + (2x * e^(x^2)) + (2x * e^(x^2)) + (2x^3 * e^(x^2))
= 2 * e^(x^2) + 4x * e^(x^2) + 2x^3 * e^(x^2)
Step 3: Compute the third derivative of f(x):
To find the third derivative, we apply the product rule again:
f'''(x) = (f''(x))' = ((2 * e^(x^2) + 4x * e^(x^2) + 2x^3 * e^(x^2)))'
= (2 * e^(x^2))' + (4x * e^(x^2))' + (2x^3 * e^(x^2))'
Now we need to differentiate each term using the chain rule:
(2 * e^(x^2))' = 2 * (e^(x^2))'
= 2 * (2x * e^(x^2))
= 4x * e^(x^2)
(4x * e^(x^2))' = 4 * e^(x^2) + 4x * (e^(x^2))'
= 4 * e^(x^2) + 4x * (2x * e^(x^2))
= 4 * e^(x^2) + 8x^2 * e^(x^2)
(2x^3 * e^(x^2))' = 2 * (3x^2) * e^(x^2) + 2x^3 * (e^(x^2))'
= 6x^2 * e^(x^2) + 2x^3 * (2x * e^(x^2))
= 6x^2 * e^(x^2) + 4x^4 * e^(x^2)
Therefore, the third derivative of f(x) = x^2 * e^(x^2) is:
f'''(x) = 4x * e^(x^2) + 4 * e^(x^2) + 8x^2 * e^(x^2) + 6x^2 * e^(x^2) + 4x^4 * e^(x^2)
= (4x + 8x^2 + 6x^2 * x^2) * e^(x^2)
= (4x + 8x^2 + 6x^4) * e^(x^2)
So, the third derivative of f(x) = x^2 * e^(x^2) is (4x + 8x^2 + 6x^4) * e^(x^2).