Question

Find the derivative of the function

15. \(g\left( x \right) = {e^{{x^2} - x}}\)

Short answer

The derivative of the function is \(g'\left( x \right) = {e^{{x^2} - x}}\left( {2x - 1} \right)\).

Step-by-step solution

The Chain Rule

For two functions, the chain rule is defined as:

\(F'\left( x \right) = f'\left( {g\left( x \right)} \right) \cdot g'\left( x \right)\)

Find the derivative of the function

Use the chain rule to obtain the derivative of the function as shown below

\(\begin{aligned} g'\left( x \right) &= \frac{d}{{dx}}\left( {{e^{{x^2} - x}}} \right)\\ &= \left( {{e^{{x^2} - x}}} \right) \cdot \frac{d}{{dx}}\left( {{x^2} - x} \right)\\ &= {e^{{x^2} - x}}\left( {2x - 1} \right)\end{aligned} \)

Thus, the derivative of the function is \(g'\left( x \right) = {e^{{x^2} - x}}\left( {2x - 1} \right)\).