Problem 19
Question
Differentiate the functions in Problems 1-28. Assume that \(A\), \(B\), and \(C\) are constants. \(P(t)=C e^{t}\).
Step-by-Step Solution
Verified Answer
The derivative of \( P(t) = C e^t \) is \( P'(t) = C \, e^t \).
1Step 1: Identify the function and its components
The function given is \( P(t) = C e^t \), where \( C \) is a constant and \( t \) is the variable.
2Step 2: Recall the derivative of the exponential function
The derivative of \( e^t \) with respect to \( t \) is \( e^t \). This is a fundamental rule of differentiation for exponential functions.
3Step 3: Apply the constant multiplication rule
When a constant is multiplied by a function, the derivative of the product is the constant multiplied by the derivative of the function. Therefore, the derivative of \( C e^t \) with respect to \( t \) is \( C \cdot \frac{d}{dt}(e^t) = C \, e^t \).
4Step 4: Write the final derivative
Thus, the derivative of \( P(t) = C e^t \) with respect to \( t \) is simply \( P'(t) = C \, e^t \).
Key Concepts
Exponential FunctionsDerivative RulesConstant Multiplication Rule
Exponential Functions
Exponential functions are mathematical functions of the form \( f(t) = a \, e^{kt} \), where \( e \) is the base of the natural logarithm (approximately 2.71828), \( a \) and \( k \) are constants, and \( t \) is the variable. These functions are used extensively in fields like finance, biology, and physics due to their property of constant relative growth rate. In simpler terms, they model processes that increase or decrease at a rate proportional to their current value.
The function in our exercise, \( P(t) = C \, e^t \), is a specific case where the exponential growth rate is 1 (since \( e^1 = e \)). This means the function increases steadily over time. Understanding exponential functions helps in solving differentiation problems like the one we are dealing with.
The function in our exercise, \( P(t) = C \, e^t \), is a specific case where the exponential growth rate is 1 (since \( e^1 = e \)). This means the function increases steadily over time. Understanding exponential functions helps in solving differentiation problems like the one we are dealing with.
- Exponential functions always have a base \( e \) which is constant.
- They represent processes that grow or decay continuously and proportionally.
- The differentiation of such functions is straightforward due to their consistent rate of change.
Derivative Rules
Derivative rules are fundamental when dealing with differentiation problems. They help us find the rate at which a function is changing at any point. Differentiation is a core operation in calculus, used to compute this rate of change. Let’s focus on one of these rules often applied to exponential functions.
The most relevant derivative rule for exponential functions, like \( e^t \), is that the derivative of \( e^t \) with respect to \( t \) is simply \( e^t \). Why? Because exponential functions have a unique property where the rate of change is equal to the value of the function itself. This property makes them especially simple to differentiate compared to other functions.
The most relevant derivative rule for exponential functions, like \( e^t \), is that the derivative of \( e^t \) with respect to \( t \) is simply \( e^t \). Why? Because exponential functions have a unique property where the rate of change is equal to the value of the function itself. This property makes them especially simple to differentiate compared to other functions.
- Knowing basic derivative rules, like the one for exponential functions, simplifies complex calculations.
- These rules are essential for swiftly solving differentiation problems in both academic and real-world scenarios.
Constant Multiplication Rule
The constant multiplication rule is crucial when differentiating functions that involve constants. It states that if you have a function \( f(t) = C \, g(t) \), where \( C \) is a constant, then the derivative of this product is \( C \) times the derivative of \( g(t) \). Understanding this rule simplifies the process of differentiation and avoids potential errors.
In our specific exercise, the function \( P(t) = C \, e^t \), employs this rule. Here, \( C \) is multiplied by \( e^t \). Given that we already established \( \frac{d}{dt}(e^t) = e^t \), applying the constant multiplication rule directly gives us: \( \frac{d}{dt}(C \, e^t) = C \, e^t \). Simple, right?
In our specific exercise, the function \( P(t) = C \, e^t \), employs this rule. Here, \( C \) is multiplied by \( e^t \). Given that we already established \( \frac{d}{dt}(e^t) = e^t \), applying the constant multiplication rule directly gives us: \( \frac{d}{dt}(C \, e^t) = C \, e^t \). Simple, right?
- This rule highlights the simplicity in differentiating functions where constants are involved.
- It avoids unnecessary calculations and focuses directly on the core operation needed.
- Always remember to multiply the constant with the derived function after applying this rule.
Other exercises in this chapter
Problem 19
Differentiate the functions in Problems 1-20. Assume that \(A\) and \(B\) are constants. $$ f(t)=\frac{t^{2}}{\cos t} $$
View solution Problem 19
Find the derivative. Assume that \(a, b, c\), and \(k\) are constants. $$ f(w)=\left(5 w^{2}+3\right) e^{w^{2}} $$
View solution Problem 19
Find the derivative. Assume \(a, b, c, k\) are constants. $$y=\frac{1}{r^{7 / 2}}$$
View solution Problem 20
Differentiate the functions in Problems 1-20. Assume that \(A\) and \(B\) are constants. $$ f(\theta)=\frac{\sin \theta}{\theta} $$
View solution