Final Answer:
The derivative of f(x) = ex xe is f'(x) = ex(xe-1) = exe-1.
Step-by-step explanation:
To find the derivative of a function, we use the power rule, which states that the derivative of x^n is nx^(n-1). However, when we have a product of ex and xe, we need to apply the product rule, which is the sum of the derivatives of each term multiplied by the original function.
First, let's find the derivative of ex. Using the power rule, we get ex^(e-1). Next, let's find the derivative of xe. Using the product rule, we get e . xe^(xe-1).
Now, let's combine these two derivatives. We have e
+ e*x
, which simplifies to exe-1. This is our final answer for the derivative of f(x) = ex xe.
In other words, as x increases, f(x) increases at an exponentially decreasing rate. This means that as we move further away from the origin, the rate at which f(x) is growing decreases exponentially. This behavior is a result of the product of two exponential functions with different bases.
As an example, let's say we have f(2) = 8 and we want to find f'(2). Using our derivative formula, we get f'(2) = e^2 . 2e^(2*2-1) = 32e³. This tells us that at x = 2, f(x) is growing at a rate of approximately 320e³ units per unit increase in x. As x continues to increase, this rate will continue to decrease exponentially due to the negative exponent in e^(-1).