The function 𝑓(𝑥)=$e^{-x^2}$models the intensity of a signal emitted by a transmitter, where 𝑥 represents the distance from the source. This type of model shows how the signal’s intensity varies based on distance, peaking near the source and decreasing further away.

• Recognize that we need to use the chain rule to differentiate $f(x) = e^{-x^2}$ 1 mark

• Apply chain rule: $f'(x) = e^{-x^2} \cdot \frac{d}{dx}(-x^2)$ 1 mark

• Final answer: $f'(x) = -2xe^{-x^2}$ 1 mark

When differentiating exponential functions with a variable exponent, remember to use the chain rule and multiply by the derivative of the exponent

• Let $f'(x) = -2xe^{-x^2}$ A1

• Recognize that $e^{-x^2}$ is always positive for all $x$ A1

• Therefore sign of $f'(x)$ depends only on $-2x$:

  • When $x < 0$: $-2x > 0$ so $f'(x) > 0$
  • When $x > 0$: $-2x < 0$ so $f'(x) < 0$

• Interval where signal intensity is increasing: $(-\infty,0)$ A1

Must include correct interval notation for final mark