Differentiating $-2e^{-5t}$ A Step-by-Step Guide

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Introduction to Differentiation

In the realm of calculus, differentiation stands as a fundamental operation, enabling us to determine the rate at which a function's output changes with respect to its input. At its core, differentiation involves finding the derivative of a function, which essentially represents the slope of the tangent line at any given point on the function's graph. This concept has far-reaching applications across various fields, including physics, engineering, economics, and computer science, where understanding rates of change is crucial for modeling and analyzing dynamic systems. For instance, in physics, differentiation helps us calculate the velocity and acceleration of a moving object, while in economics, it aids in determining marginal cost and revenue. The process of differentiation relies on a set of well-defined rules and techniques, each tailored to different types of functions. From simple polynomial functions to more complex exponential and trigonometric functions, the principles of differentiation provide a powerful toolkit for exploring the behavior of mathematical expressions. This article delves into the specifics of differentiating the function $-2e^{-5t}$, illustrating the application of these rules in a step-by-step manner. We will break down the function, identify the relevant differentiation rules, and apply them systematically to arrive at the derivative. This example not only showcases the mechanics of differentiation but also highlights the importance of understanding the underlying principles for successful problem-solving in calculus. Whether you are a student embarking on your calculus journey or a professional seeking to refresh your knowledge, this guide aims to provide a clear and concise explanation of how to differentiate exponential functions with constant coefficients. By the end of this discussion, you will have a solid understanding of the process and be well-equipped to tackle similar differentiation problems.

Understanding the Function $-2e^{-5t}$

The function $-2e^-5t}$ is a prime example of an exponential function with a constant coefficient. To effectively differentiate this function, it's crucial to first dissect its components and understand their individual roles. The function comprises several key elements a constant coefficient (-2), the exponential base (e), and an exponent that is a linear function of the variable t (-5t). The constant coefficient -2 acts as a vertical stretch and reflection of the exponential function. It scales the entire function by a factor of 2 and flips it across the x-axis due to the negative sign. This means that the function's values will be twice as large in magnitude compared to the basic exponential function $e^{-5t$, and they will be negative instead of positive. The exponential base e, known as Euler's number, is an irrational number approximately equal to 2.71828. It is the base of the natural logarithm and plays a fundamental role in many areas of mathematics and science. Exponential functions with base e exhibit unique properties that make them particularly amenable to calculus operations. The exponent -5t is a linear function of the variable t. The coefficient -5 in the exponent affects the rate of growth or decay of the exponential function. A negative coefficient indicates exponential decay, meaning the function's value decreases as t increases. The magnitude of the coefficient determines the rate of decay; in this case, -5 implies a relatively rapid decay compared to, say, -1. Understanding these components is essential because each influences how we approach the differentiation process. The constant coefficient can be factored out, simplifying the derivative calculation. The exponential function's derivative has a specific form that we will utilize. The linear exponent requires the application of the chain rule, a fundamental technique in calculus for differentiating composite functions. By breaking down the function into its constituent parts, we can strategically apply the appropriate differentiation rules and techniques, ensuring an accurate and efficient calculation of the derivative. This foundational understanding sets the stage for the subsequent steps in the differentiation process.

Rules and Techniques for Differentiation

To differentiate the function $-2e^-5t}$, we need to employ specific rules and techniques from calculus. The two primary rules that come into play here are the constant multiple rule and the chain rule. The constant multiple rule states that the derivative of a constant multiplied by a function is equal to the constant multiplied by the derivative of the function. Mathematically, this is expressed as $\frac{ddx}[cf(x)] = c \frac{d}{dx}[f(x)]$, where c is a constant and f(x) is a differentiable function. In our case, the constant multiple is -2, which we can factor out before differentiating the exponential part of the function. This simplifies the calculation by allowing us to focus on the derivative of $e^{-5t}$ first and then multiply the result by -2. The chain rule is a fundamental technique for differentiating composite functions, which are functions within functions. It states that the derivative of a composite function f(g(x)) is equal to the derivative of the outer function f evaluated at the inner function g(x), multiplied by the derivative of the inner function g'(x). Mathematically, this is expressed as $\frac{d{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$. In our function $-2e^{-5t}$, the exponential function $e^u$ acts as the outer function, and the linear function -5t acts as the inner function. To apply the chain rule, we first differentiate the outer function with respect to the inner function, which gives us $e^{-5t}$. Then, we differentiate the inner function -5t with respect to t, which gives us -5. Finally, we multiply these two results together to obtain the derivative of the composite function. In addition to these rules, it's essential to know the derivative of the basic exponential function $e^x$, which is simply $e^x$. This serves as the foundation for differentiating more complex exponential functions. By mastering these rules and techniques, you can confidently tackle a wide range of differentiation problems, including those involving exponential functions, constant multiples, and composite functions. The ability to apply these rules systematically and accurately is a cornerstone of calculus proficiency.

Step-by-Step Differentiation of $-2e^{-5t}$

Now, let's apply the rules and techniques discussed to differentiate the function $-2e^{-5t}$ step-by-step. This process will illustrate how the constant multiple rule and the chain rule work in tandem to find the derivative of this exponential function.

Step 1: Apply the Constant Multiple Rule

The constant multiple rule allows us to factor out the constant -2 from the differentiation process. This simplifies the problem by focusing on the exponential part of the function first. So, we can write:

ddt[−2e−5t]=−2ddt[e−5t]\frac{d}{dt}[-2e^{-5t}] = -2 \frac{d}{dt}[e^{-5t}]

This step separates the constant from the function, making the subsequent differentiation steps cleaner and easier to manage. We now need to find the derivative of $e^{-5t}$ and then multiply the result by -2.

Step 2: Apply the Chain Rule

The function $e^{-5t}$ is a composite function, where the outer function is $e^u$ and the inner function is u = -5t. The chain rule dictates that we differentiate the outer function with respect to the inner function and then multiply by the derivative of the inner function with respect to t. First, let's find the derivative of the outer function $e^u$ with respect to u:

ddu[eu]=eu\frac{d}{du}[e^u] = e^u

Next, we find the derivative of the inner function u = -5t with respect to t:

dudt=ddt[−5t]=−5\frac{du}{dt} = \frac{d}{dt}[-5t] = -5

Now, we apply the chain rule by multiplying these two derivatives together:

ddt[e−5t]=e−5t⋅(−5)=−5e−5t\frac{d}{dt}[e^{-5t}] = e^{-5t} \cdot (-5) = -5e^{-5t}

Step 3: Combine the Results

Finally, we multiply the result from Step 2 by the constant -2 that we factored out in Step 1:

−2ddt[e−5t]=−2(−5e−5t)=10e−5t-2 \frac{d}{dt}[e^{-5t}] = -2(-5e^{-5t}) = 10e^{-5t}

Therefore, the derivative of $-2e^{-5t}$ with respect to t is $10e^{-5t}$. This step-by-step breakdown demonstrates how the constant multiple rule and the chain rule are applied sequentially to differentiate the given function. Each step is crucial for arriving at the correct derivative, and understanding the logic behind each rule is key to mastering differentiation.

Result: The Derivative of $-2e^{-5t}$

After meticulously applying the constant multiple rule and the chain rule, we have arrived at the derivative of the function $-2e^{-5t}$. The process involved factoring out the constant -2, recognizing the composite nature of the exponential function, and applying the chain rule to differentiate the exponential term. By differentiating the outer function $e^u$ with respect to u and the inner function u = -5t with respect to t, we were able to combine these results to find the derivative of $e^{-5t}$. Finally, multiplying this derivative by the constant -2 yielded the complete derivative of the original function. Therefore, the derivative of $-2e^{-5t}$ with respect to t is:

ddt[−2e−5t]=10e−5t\frac{d}{dt}[-2e^{-5t}] = 10e^{-5t}

This result, $10e^{-5t}$, represents the instantaneous rate of change of the function $-2e^{-5t}$ at any given value of t. The positive coefficient 10 indicates that the derivative is positive, meaning the function is increasing as t increases. However, the exponential term $e^{-5t}$ implies that this increase diminishes as t becomes larger, since $e^{-5t}$ approaches 0 as t approaches infinity. This derivative provides valuable information about the behavior of the original function. It allows us to analyze the function's slope at any point, determine intervals of increasing or decreasing behavior, and identify any critical points where the function's rate of change is zero. In various applications, this information can be crucial for optimization problems, modeling dynamic systems, and understanding the function's overall characteristics. The result also reinforces the importance of mastering differentiation rules and techniques. The chain rule, in particular, is a fundamental tool for differentiating composite functions, and its correct application is essential for obtaining accurate results. By understanding and applying these rules, you can confidently tackle a wide range of differentiation problems in calculus and related fields.

Conclusion

In conclusion, differentiating the function $-2e^{-5t}$ provides a clear demonstration of the power and utility of calculus principles. Through a step-by-step application of the constant multiple rule and the chain rule, we successfully determined the derivative to be $10e^{-5t}$. This process not only illustrates the mechanics of differentiation but also highlights the importance of understanding the underlying rules and techniques. The constant multiple rule allowed us to simplify the problem by factoring out the constant -2, while the chain rule enabled us to differentiate the composite exponential function effectively. Each step in the process was crucial for arriving at the correct result, and the final derivative provides valuable insights into the behavior of the original function. The derivative $10e^{-5t}$ represents the instantaneous rate of change of $-2e^{-5t}$ and can be used to analyze its slope, intervals of increase or decrease, and critical points. This information is essential in various applications, including optimization problems, modeling dynamic systems, and understanding the function's characteristics. Mastering differentiation is a cornerstone of calculus, and this example showcases the importance of understanding and applying the fundamental rules and techniques. The chain rule, in particular, is a versatile tool for differentiating composite functions, and its correct application is crucial for obtaining accurate results. By developing a strong foundation in differentiation, you can confidently tackle a wide range of problems in mathematics, physics, engineering, economics, and other fields. The ability to differentiate functions accurately and efficiently is a valuable skill that will serve you well in your academic and professional endeavors. This comprehensive guide has aimed to provide a clear and concise explanation of how to differentiate exponential functions with constant coefficients, equipping you with the knowledge and skills to approach similar problems with confidence.