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Chapter 4: Problem 24

Simplify the function before differentiating. $$f(t)=e^{3 t}\left(e^{2 t}-e^{4 t}\right)$$

Short Answer

Expert verified
Step 1: Simplify to \(f(t) = e^{5t} - e^{7t}\). Step 2: Differentiate to get \(f'(t) = 5e^{5t} - 7e^{7t}\).

Step by step solution

01

Expand the Function

First, distribute the exponent term. We have: \[ f(t) = e^{3t} \times (e^{2t} - e^{4t}) \]This simplifies to:\[ f(t) = e^{3t + 2t} - e^{3t + 4t} \]Hence,\[ f(t) = e^{5t} - e^{7t} \]
02

Differentiate the Function

Now, differentiate the simplified function with respect to \( t \). Recall that \( \frac{d}{dt} e^{kt} = ke^{kt} \). So, we calculate:\[ f'(t) = \frac{d}{dt} (e^{5t}) - \frac{d}{dt} (e^{7t}) \]This gives:\[ f'(t) = 5e^{5t} - 7e^{7t} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Function Simplification
Simplifying functions is a crucial first step before applying differentiation. In this exercise, the function given is: \(f(t) = e^{3t}(e^{2t} - e^{4t})\)First, we distribute the exponential term, helping us break down and simplify the initial function. By distributing \(e^{3t}\), we perform the arithmetic inside the exponents:
  • The original function becomes \(e^{3t} \times (e^{2t} - e^{4t})\)
  • Next, applying the laws of exponents, we rewrite this as \(e^{3t + 2t} - e^{3t + 4t}\)
  • This results in the simplified form \(e^{5t} - e^{7t}\)
In essence, simplifying the function makes the differentiation process much more manageable.
Exponential Functions
Exponential functions are functions where the variable is in the exponent. In this article, we are focusing on expressions like \(e^{kt}\), where \(e\) is the base of the natural logarithm, and \(k\) is a constant.
  • The laws of exponents state that \(a^{m} \times a^{n} = a^{m+n}\). This helps in breaking down complex exponents while simplifying functions.
  • For instance, \(e^{3t} \times e^{2t}\) becomes \(e^{3t+2t} = e^{5t}\) because the bases are the same (both are \(e\)).
  • The initial simplification step converts a complicated function into a simpler form, allowing more straightforward differentiation.
Understanding these rules helps immensely, especially when you face mixed exponential terms.
Calculus
Calculus involves the study of rates of change and accumulation. Differentiation, a core concept of calculus, measures how a function changes as its input changes. Here, we need to differentiate the simplified form:\(f(t) = e^{5t} - e^{7t}\)To differentiate exponential functions, we use the rule \( \frac{d}{dt} e^{kt} = ke^{kt} \).
  • For \(e^{5t}\), applying the rule, we get \(5e^{5t}\).
  • Similarly, for \(e^{7t}\), we get \(7e^{7t}\).
  • Subtracting these results gives us\( f'(t) = 5e^{5t} - 7e^{7t}\).
This step-by-step approach shows the importance of simplification and deep understanding of exponential differentiation rules within calculus.

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Most popular questions from this chapter

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