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} )