Question

Find the derivative of the exponential function:
e^-x^2

Answer 1

\[e ^{-x ^{2}}\]

Answer 2

-2xe^(-x^2)

Answer 3

\[-2\times x \times \exp ^{-x^2}\]

Answer 4

The second and third responses are both correct. Here's why:
\[y=e ^{-x ^{2}}\]
is what you gave; if you set a new variable \[u=(-x)^{2}\]
Then it's relatively easy to find du/dx:
\[du/dx=2(-x)^{2-1}=2(-x)=-2x\]
This will come in handy later.
Now, if we take the original equation and differentiate it in terms of x, we get
\[dy/dx=d/dx(e ^{-x ^{2}})=d/dx(e ^{u})\]
By the Chain Rule, we get
\[dy/dx=d/dx(e ^{u})=d/du(e ^{u})du/dx\]
By differentiating in terms of u, we get
\[dy/dx=e ^{u}du/dx\]
If we then substitute u and du/dx for the equations we stated above, we get
\[dy/dx=e ^{-x ^{^{2}}}(-2x)=-2xe ^{-x ^{2}}\]