Question

Compute the derivative of the following functions. $$f(x)=15 e^{3 x}$$

Short answer

Answer: The derivative of the function is \(f'(x) = 45e^{3x}\).

Step-by-step solution

Identify the derivative rule needed to be applied

For the given function, \(f(x)=15 e^{3x}\), we need to apply the exponential function derivative rule, which states: If \(f(x)=ae^{kx}\), then \(f'(x)=ake^{kx}\). In our case, \(a=15\) and \(k=3\).

Apply the derivative rule to find the derivative

Applying the exponential derivative rule to our function, we have: \(f'(x) = 15 \cdot 3 e^{3x}\)

Simplify the expression

Now, we simplify the expression: \(f'(x) = 45 e^{3x}\) So, the derivative of the given function is: $$f'(x) = 45 e^{3x}$$

Key concepts

Calculus

Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. It's pivotal for analyzing and understanding change in various contexts, such as physical sciences, computer sciences, economics, business, medicine, demographics, and in many other fields where a quantitative description of systems is required. In our exercise, calculus helps to find the rate at which a quantity changes, which in this case is the derivative of an exponential function.

Derivative Rules

In calculus, derivative rules are formulas that provide a method to find the rate at which a function is changing at any given point. The main rules include the power rule, product rule, quotient rule, and chain rule. For exponential functions like \(e^{kx}\), the derivative is quite straightforward; if \(f(x) = ae^{kx}\), the derivative is \(f'(x) = ake^{kx}\). This rule is essential for handling functions involving exponential terms, which commonly occur in modeling exponential growth or decay.

Exponential Growth

Exponential growth describes a process where the quantity grows at a rate proportional to its current value. This concept often applies to populations, investments, and infectious diseases spread, showcasing how quickly things can escalate or increase. The function \(f(x) = ae^{kx}\) represents exponential growth when \(k > 0\), and the derivative \(f'(x) = ake^{kx}\) illustrates how the rate of change is also exponential. In the given exercise, the base of the exponential function is Euler's number \(e\), which is approximately 2.71828, indicating a naturally occurring growth rate.

Chain Rule

The chain rule is a derivative rule used to compute the derivative of the composition of two or more functions. It states that if \(h(x) = f(g(x))\), then the derivative of \(h\) with respect to \(x\) is \(h'(x) = f'(g(x)) \times g'(x)\). It's akin to peeling an onion, working from the outside function to the inside function. In the scenario of our provided exercise, the function \(f(x) = 15e^{3x}\) could also be approached with the chain rule by identifying the outer function as \(15e^u\) (where \(u = 3x\)) and the inner function as \(u = 3x\), and then applying the derivative rules respectively.