How do you differentiate f(x) = x^(5/ 2) e^x?

Hi there! To differentiate a function in the form of $f \left(x\right) = g \left(x\right) \cdot h \left(x\right)$, you want to use the product rule.

Explanation:

To start, the general form of the derivative of a product of two functions is:

$f ' \left(x\right) = g ' \left(x\right) \cdot h \left(x\right) + h ' \left(x\right) \cdot g \left(x\right)$

It's important to note that the order of the multiplication doesn't matter as multiplication is commutative whereby $5 \cdot 4 = 4 \cdot 5$.

Let's get our derivatives of each part of the function first:

$g ' \left(x\right) \to$ For this we use power rule so:

$g ' \left(x\right) = \frac{5}{2} {x}^{\frac{3}{2}}$

$h ' \left(x\right) \to$ For this, the derivative of ${e}^{x}$ is always ${e}^{x}$.

Putting it all together we get:

$f ' \left(x\right) = \frac{5}{2} {x}^{\frac{3}{2}} \cdot {e}^{x} + {e}^{x} \cdot {x}^{\frac{5}{2}}$

And that's it! To remember the general form for product rule, you can think "derivative of the first, times the second, plus derivative of the second times the first", that's how I've always remembered it!

Hopefully everything was clear! If you have any questions, feel free to ask! :)