Question

find the derivative of ( h(x) = e^{x^{2}-3x} ).

Explanation:

Step1: Let \( u = x^{2}-3x \)

So \( h(x)=e^{u} \)

Step2: Find \( \frac{du}{dx} \)

Using power rule: \( \frac{du}{dx}=\frac{d}{dx}(x^{2}-3x)=2x - 3 \)

Step3: Find \( \frac{dh}{du} \)

Since \( h(u)=e^{u} \), then \( \frac{dh}{du}=e^{u} \)

Step4: Apply chain rule \( \frac{dh}{dx}=\frac{dh}{du}\cdot\frac{du}{dx} \)

Substitute \( u = x^{2}-3x \), \( \frac{dh}{du}=e^{u} \) and \( \frac{du}{dx}=2x - 3 \)
\( \frac{dh}{dx}=e^{x^{2}-3x}\cdot(2x - 3)=(2x - 3)e^{x^{2}-3x} \)

Answer:

\( h^{\prime}(x)=(2x - 3)e^{x^{2}-3x} \)