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Chapter 3: Problem 12

\(3-32\) Differentiate the function. \(y=5 e^{x}+3\)

Short Answer

Expert verified
The derivative is \( 5e^x \).

Step by step solution

01

Identify the Function Type

The given function is a sum of an exponential function and a constant: \( y = 5e^x + 3 \). We need to differentiate each part separately.
02

Differentiate the Exponential Term

Differentiate the term \( 5e^x \). The derivative of \( e^x \) is \( e^x \). Using the constant multiple rule, the derivative of \( 5e^x \) is \( 5e^x \).
03

Differentiate the Constant Term

Differentiate the constant term \( 3 \). The derivative of any constant is 0.
04

Combine the Derivatives

Combine the results from the previous steps to get the derivative of the entire function. So, \( \frac{dy}{dx} = 5e^x + 0 = 5e^x \).

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Key Concepts

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

Exponential Functions
An exponential function is a type of mathematical function in the form \(f(x) = a \cdot e^{bx}\), where \(e\) is the base of the natural logarithm, often called Euler's number, and it approximately equals 2.71828. Exponential functions grow at a rate proportional to their value, which makes them incredibly important in models of growth and decay.
This characteristic also defines their derivatives. For example, the derivative of \(e^x\) with respect to \(x\) is \(e^x\), meaning it remains the same. This behavior is unique to exponential functions, as it preserves the function's form after differentiation. This makes it straightforward to work with in calculus.
  • The base \(e\) ensures that the function will grow exponentially, meaning very rapidly, or decay depending on the coefficients and powers involved.
  • In the given exercise, the exponential function is \(5e^x\), where 5 is a constant multiple and \(x\) is the exponent of \(e\).
Constant Rule
The constant rule is one of the simplest rules in differentiation. It states that the derivative of a constant function is zero. This is because a constant does not change, regardless of the variable value. Thus, its rate of change, which is what the derivative measures, is zero.
In the context of the exercise, the term \(3\) is a constant. When you differentiate \(3\), the result is 0 because constants do not change and thus have no rate of change.
  • This rule is crucial when dealing with polynomials or any function with an added constant, as it simplifies the differentiation process.
  • No matter what the constant is, be it \(3\), \(7\), or any other number, its derivative will always be zero.
Derivative
A derivative, in mathematical terms, represents the rate of change of a function with respect to a variable. It's like getting the slope at any point for a function graph. Differentiation is the operation used to find a derivative.
The process involves applying rules such as the power rule, product rule, quotient rule, and chain rule, depending on the structure of the function.
  • In simple terms, if you imagine a curve following a path, finding the derivative is like measuring at what rate you're climbing or descending as you move along that path.
  • In our exercise, the derivative \(\frac{dy}{dx}\) was obtained by differentiating the function \(y = 5e^x + 3\), resulting in \(5e^x\).
Constant Multiple Rule
The constant multiple rule simplifies differentiation when a constant is multiplied by a function. It states that the derivative of a constant times a function is the constant times the derivative of the function.
Consider \(f(x) = a \cdot g(x)\), where \(a\) is a constant, and \(g(x)\) is a differentiable function. The derivative is \(a \cdot g'(x)\).
  • This means in the exercise, the term \(5e^x\) becomes \(5 \cdot \frac{d}{dx}[e^x] = 5e^x\), as the derivative of \(e^x\) is \(e^x\).
  • This rule is extremely useful because it allows us to "factor out" constants, so to speak, making the process of differentiation more straightforward.
  • By using this rule, it becomes clear how to tackle compound functions involving constants and simplifies finding derivatives with minimal computation.

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Most popular questions from this chapter

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