Problem 2 $$3 t^{2}-e^{2 t}$$... [FREE SOLUTION]

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Fundamentals of Differential Equations

R. Kent Nagle

$Math Studyset 91影视 Explanations$ Math

9 Edition

Chapter 7: Problem 2

$$3 t^{2}-e^{2 t}$$

Short Answer

Expert verified

The derivative of the function $ 3 t^{2}-e^{2 t} $ with respect to $ t $ is $ 6t - 2e^{2t} $

Step by step solution

Differentiate the polynomial term

Differentiate the term $ 3 t^{2} $ using power rule. Which gives $ 2*3*t^{1} = 6t $.

Differentiate the exponential term

Now differentiate the term $ -e^{2 t} $ using chain rule. First, differentiate the outer function, which is $ e^u $, where $ u = 2t $. The derivative of $ e^u $ is simply $ e^u $. So, we get $ e^{2 t} $. Second, differentiate the inner function $ 2t $, which gives us 2. By chain rule, we multiply these two results. So, we get $ -2e^{2t} $.

Write out the final derivative by combining both terms

After finding the derivatives of both terms, we get our final derivative as $ 6t - 2e^{2t} $

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Key Concepts

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

Chain Rule

The Chain Rule is a fundamental method in differential calculus used to differentiate composite functions.
When a function is composed of two or more functions, we need the chain rule to find its derivative. For example, consider the function $-e^{2t}$. In this case, we have an exponential function $e^u$, where $u = 2t$.
The outer function is $e^u$, and the inner function is $u=2t$. The chain rule tells us to:

First, differentiate the outer function with respect to the inner function: the derivative of $e^u$ is still $e^u$. Thus, when applying it to $e^{2t}$, it remains $e^{2t}$.
Next, differentiate the inner function $2t$ with respect to $t$, which is simply 2.
Finally, multiply these derivatives together: $-2 \times e^{2t}$, resulting in $-2e^{2t}$.

Hence, by using the chain rule, we systematically tackle composite functions by breaking them into simpler parts.

Exponential Functions

Exponential functions are a class of functions where the variable is in the exponent, such as $e^{x}$.
The base $e$ is a mathematical constant approximately equal to 2.71828, and it has a unique property where the derivative of $e^x$ is also $e^x$. This makes it particularly simple and consistent in calculus.
For the given problem, we have $-e^{2t}$, an example of an exponential function where the argument (the exponent) itself contains a function of $t$.
When differentiating an exponential function like $e^{u}$, we

First retain the base expression: differentiate $e^u$ with respect to $u$ to get $e^u$.
Second, apply the chain rule if $u$ is a function of another variable.

This characteristic of employing the chain rule when the exponent itself is a function is essential for dealing with exponential terms neatly.

Power Rule

The Power Rule is one of the simplest and most frequently used rules in differential calculus when it comes to differentiating polynomials.
This rule states that for any term of the form $at^n$, where $a$ and $n$ are constants, its derivative is $n \cdot at^{n-1}$.

For example, to differentiate the term $3t^2$, we apply the power rule.
The current power of $t$ is 2, so we multiply 2 by the coefficient 3 to get 6.
The new power of $t$ becomes $2-1 = 1$, resulting in the term $6t$.

This transforms a polynomial term into its derivative in a straightforward manner. The power rule simplifies the process of handling polynomial functions significantly, contributing to swift calculus computations.

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91影视

$$3 t^{2}-e^{2 t}$$

Short Answer

Step by step solution

Differentiate the polynomial term

Differentiate the exponential term

Write out the final derivative by combining both terms

Key Concepts

Chain Rule

Exponential Functions

Power Rule

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