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Exponential functions are mathematical functions of the form \(f(x) = a e^{bx}\), where \(e\) is the base of the natural logarithm, and \(a\) and \(b\) are constants. They are often found in real-world phenomena such as population growth, radioactive decay, and interest calculations. The key property of exponential functions is that their growth rate is proportional to their current value.
To differentiate an exponential function, you need to understand that the derivative of \(e^x\) is \(e^x\) itself. Thus, for a general exponential function \(ae^{bx}\), the derivative is \(abe^{bx}\).
In the given problem, we have the term \(3e^x\), where the constant \(a = 3\) and \(b = 1\). According to our rule, the derivative is \(3e^x\).

Linear functions are one of the simplest types of functions, with the general form \(f(x) = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept. These functions graph as straight lines and are foundational in understanding calculus and higher-level math concepts.
The derivative of a linear function is straightforward. It’s essentially the slope \(m\) of the function, since the rate of change is constant.
For the linear term \(-7x\) in our given function, the slope is \(-7\). Therefore, the derivative of this term is simply \(-7\).

Step-by-step solutions are incredibly valuable in understanding complex problems. They provide a structured method to break down a problem into manageable parts. Let’s revisit the steps to differentiate \(y = 3e^x - 7x\):

• Step 1: Differentiate the exponential function \(3e^x\). As mentioned, its derivative is \(3e^x\).
• Step 2: Differentiate the linear term \(-7x\). Its derivative is \(-7\).
• Step 3: Combine the derivatives from Steps 1 and 2 to get the final result: \(3e^x - 7\).

Breaking down problems into these smaller steps helps to ensure that no detail is overlooked and builds a clear path to the solution.
It’s important to practice solving problems step-by-step to build confidence and accuracy in your mathematical abilities.