Find f′(x) if f(x) = 15e^3x.

Verified step by step guidance

1. Step 1: Identify the function f(x) = 15e^{3x}. This is an exponential function where the base is e and the exponent is 3x.
2. Step 2: Recall the derivative rule for exponential functions: if f(x) = e^{u(x)}, then f'(x) = u'(x) e^{u(x)}.
3. Step 3: Identify u(x) in the function. Here, u(x) = 3x.
4. Step 4: Differentiate u(x) with respect to x. The derivative of u(x) = 3x is u'(x) = 3.
5. Step 5: Apply the derivative rule: f'(x) = u'(x) e^{u(x)}. Substitute u'(x) = 3 and u(x) = 3x into the formula to find f'(x).

Key Concepts

Derivative

The derivative of a function measures how the function's output changes as its input changes. It is a fundamental concept in calculus that provides the slope of the tangent line to the function's graph at any given point. The notation f′(x) represents the derivative of the function f(x) with respect to x.

Exponential Functions

Exponential functions are mathematical functions of the form f(x) = a * e^(bx), where e is Euler's number (approximately 2.71828), and a and b are constants. These functions are characterized by their rapid growth or decay and are commonly encountered in calculus. The derivative of an exponential function is proportional to the function itself, which simplifies differentiation.

Chain Rule

The chain rule is a fundamental technique in calculus used to differentiate composite functions. It states that if a function y = f(g(x)) is composed of two functions, the derivative can be found by multiplying the derivative of the outer function f with the derivative of the inner function g. This rule is particularly useful when dealing with functions that involve exponentials, as seen in the given problem.