Consider The Function \[$ F(x) = 3^x + X \$\].1. Applying Differentiation Rules (Section 3.3) - Use The Appropriate Rules (including The Chain Rule) To Find The Derivative \[$ F^{\prime}(x) \$\]. - When Evaluating \[$

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Differentiation of the Function f(x) = 3^x + x

In this section, we will apply the differentiation rules to find the derivative of the function f(x) = 3^x + x. The function f(x) is a combination of two functions, 3^x and x, and we will use the appropriate rules, including the chain rule, to find its derivative.

To find the derivative of f(x) = 3^x + x, we will apply the following differentiation rules:

  • Power Rule: If f(x) = x^n, then f'(x) = nx^(n-1)
  • Chain Rule: If f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x)
  • Sum Rule: If f(x) = g(x) + h(x), then f'(x) = g'(x) + h'(x)

Step 1: Differentiate 3^x

Using the Power Rule, we can differentiate 3^x as follows:

f(x) = 3^x f'(x) = 3^x * ln(3)

where ln(3) is the natural logarithm of 3.

Step 2: Differentiate x

Using the Power Rule, we can differentiate x as follows:

f(x) = x f'(x) = 1

Step 3: Apply the Chain Rule

Since f(x) = 3^x + x is a combination of two functions, we will apply the Chain Rule to find its derivative. Let's define g(x) = 3^x and h(x) = x. Then, f(x) = g(h(x)).

Using the Chain Rule, we can find the derivative of f(x) as follows:

f'(x) = g'(h(x)) * h'(x) = (3^x * ln(3)) * 1 = 3^x * ln(3) + 1

Step 4: Simplify the Derivative

We can simplify the derivative of f(x) by combining the two terms:

f'(x) = 3^x * ln(3) + 1

In this section, we applied the differentiation rules to find the derivative of the function f(x) = 3^x + x. We used the Power Rule, Chain Rule, and Sum Rule to find the derivative of f(x), which is f'(x) = 3^x * ln(3) + 1.

Let's find the derivative of f(x) = 3^x + x at x = 2.

f'(x) = 3^x * ln(3) + 1 f'(2) = 3^2 * ln(3) + 1 = 9 * ln(3) + 1

The derivative of f(x) = 3^x + x is a combination of two terms: 3^x * ln(3) and 1. The first term represents the derivative of 3^x, while the second term represents the derivative of x.

The derivative of f(x) = 3^x + x is an important concept in calculus, as it allows us to find the rate of change of the function f(x) with respect to x.

The derivative of f(x) = 3^x + x has many applications in mathematics and science. For example, it can be used to find the maximum or minimum value of a function, or to model the behavior of a physical system.

In this section, we applied the differentiation rules to find the derivative of the function f(x) = 3^x + x. We used the Power Rule, Chain Rule, and Sum Rule to find the derivative of f(x), which is f'(x) = 3^x * ln(3) + 1. The derivative of f(x) = 3^x + x is an important concept in calculus, and it has many applications in mathematics and science.
Q&A: Differentiation of the Function f(x) = 3^x + x

In our previous article, we applied the differentiation rules to find the derivative of the function f(x) = 3^x + x. In this article, we will answer some common questions related to the differentiation of this function.

Q: What is the derivative of f(x) = 3^x + x?

A: The derivative of f(x) = 3^x + x is f'(x) = 3^x * ln(3) + 1.

Q: How do I find the derivative of 3^x?

A: To find the derivative of 3^x, you can use the Power Rule, which states that if f(x) = x^n, then f'(x) = nx^(n-1). In this case, n = 1, so f'(x) = 3^x * ln(3).

Q: How do I find the derivative of x?

A: To find the derivative of x, you can use the Power Rule, which states that if f(x) = x^n, then f'(x) = nx^(n-1). In this case, n = 1, so f'(x) = 1.

Q: What is the Chain Rule?

A: The Chain Rule is a differentiation rule that states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x). This rule is used to find the derivative of composite functions.

Q: How do I apply the Chain Rule to find the derivative of f(x) = 3^x + x?

A: To apply the Chain Rule, you need to define two functions, g(x) = 3^x and h(x) = x. Then, you can find the derivative of f(x) = g(h(x)) using the Chain Rule: f'(x) = g'(h(x)) * h'(x) = (3^x * ln(3)) * 1 = 3^x * ln(3) + 1.

Q: What are some common applications of the derivative of f(x) = 3^x + x?

A: The derivative of f(x) = 3^x + x has many applications in mathematics and science, including:

  • Finding the maximum or minimum value of a function
  • Modeling the behavior of a physical system
  • Calculating the rate of change of a function

Q: How do I use the derivative of f(x) = 3^x + x to find the maximum or minimum value of a function?

A: To use the derivative of f(x) = 3^x + x to find the maximum or minimum value of a function, you need to find the critical points of the function by setting the derivative equal to zero and solving for x. Then, you can use the second derivative test to determine whether the critical point is a maximum or minimum.

Q: What is the second derivative test?

A: The second derivative test is a test used to determine whether a critical point is a maximum or minimum. If the second derivative is positive at a critical point, then the critical point is a minimum. If the second derivative is negative at a critical point, then the critical point is a maximum.

In this article, we answered some common questions related to the differentiation of the function f(x) = 3^x + x. We also discussed some common applications of the derivative of this function, including finding the maximum or minimum value of a function and modeling the behavior of a physical system.