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Exponential functions are mathematical expressions where a constant base is raised to a power that includes a variable. A very common base for exponentials in calculus is the constant \(e\), approximately equal to 2.718. These functions are incredibly useful in modeling real-world phenomena such as population growth, radioactive decay, and interest calculations.

An exponential function generally takes the form \(a \, e^{g(x)}\), where \(a\) is a constant multiplier, \(e\) is the base, and \(g(x)\) is an expression in \(x\). In our specific problem, the exponential function is \(f(x) = 3 e^{2-5x}\). Here, \(a = 3\) and \(g(x) = 2 - 5x\).

Key characteristics of exponential functions include:

• The rate of change, or growth, increases or decreases rapidly based on the sign and coefficient of the exponent \(x\).
• Exponential functions never touch the x-axis; they asymptotically approach it as \(x\) moves towards infinity or negative infinity.
• If the coefficient of \(x\) in the exponent is negative, the function represents exponential decay, while a positive coefficient signifies exponential growth.

Derivative rules are the guidelines used to calculate the derivative of a function. The derivative represents the rate at which a function changes, often thought of as the slope of the tangent line at any given point on the curve.

Several rules apply when differentiating functions:

• Constant Rule: The derivative of a constant is zero.
• Power Rule: For any function \(x^n\), the derivative is \(nx^{n-1}\).
• Product Rule: If you have a product of two functions, say \(u(x) and v(x)\), then the derivative is \(u'v + uv'\).
• Chain Rule: For composite functions, the rule states that the derivative of \(f(g(x))\) is \(f'(g(x))g'(x)\).

In our exercise, the chain rule is particularly relevant. Since \(f(x) = 3e^{2-5x}\), we first differentiate \(g(x) = 2 - 5x\) to obtain \(g'(x) = -5\), then apply the rule for differentiating \(e^{g(x)}\) as \(g'(x) e^{g(x)}\). This approach allows us to handle complex exponential functions efficiently.

Differentiating a linear function is one of the simplest tasks in calculus. A linear function is generally expressed in the form \(ax + b\), where \(a\) and \(b\) are constants, and \(x\) is the variable. The derivative of a linear function \(ax + b\) is simply the coefficient \(a\).

In the context of our problem, the function to differentiate within the exponential is \(g(x) = 2 - 5x\). This is a linear function where \(a = -5\) and \(b = 2\). Thus, the differentiation process is straightforward:

• The derivative of the constant \(2\) is \(0\).
• The derivative of \(-5x\) is \(-5\).

Therefore, the derivative \(g'(x)\) is \(-5\). This result is crucial in applying the chain rule for calculating the derivative of the entire exponential function \(f(x) = 3e^{2-5x}\). Recognizing and understanding these basic differentiation techniques is a fundamental skill in calculus.