Find \[$ F^{\prime}(x) \$\].Given:$\[ F(x) = (3x + 7)(2x - 6) \\]$\[ F^{\prime}(x) = \square \\](Type An Exact Answer.)

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Introduction

In calculus, the derivative of a function represents the rate of change of the function with respect to its input. When dealing with a product of functions, we can use the product rule to find the derivative. In this article, we will explore how to find the derivative of a product of two functions using the product rule.

The Product Rule

The product rule is a fundamental concept in calculus that allows us to find the derivative of a product of two functions. The product rule states that if we have two functions, f(x) and g(x), then the derivative of their product is given by:

f′(x)g(x)+f(x)g′(x){ f^{\prime}(x)g(x) + f(x)g^{\prime}(x) }

Applying the Product Rule to the Given Function

Given the function:

f(x)=(3x+7)(2x−6){ f(x) = (3x + 7)(2x - 6) }

We can use the product rule to find the derivative of this function. To do this, we need to find the derivatives of the two functions that make up the product.

Finding the Derivatives of the Individual Functions

First, let's find the derivative of the first function, 3x + 7.

ddx(3x+7)=3{ \frac{d}{dx}(3x + 7) = 3 }

Next, let's find the derivative of the second function, 2x - 6.

ddx(2x−6)=2{ \frac{d}{dx}(2x - 6) = 2 }

Using the Product Rule to Find the Derivative

Now that we have the derivatives of the individual functions, we can use the product rule to find the derivative of the product.

f′(x)=(3)(2x−6)+(3x+7)(2){ f^{\prime}(x) = (3)(2x - 6) + (3x + 7)(2) }

Simplifying the Derivative

To simplify the derivative, we can distribute the terms and combine like terms.

f′(x)=6x−18+6x+14{ f^{\prime}(x) = 6x - 18 + 6x + 14 }

f′(x)=12x−4{ f^{\prime}(x) = 12x - 4 }

Conclusion

In this article, we used the product rule to find the derivative of a product of two functions. We first found the derivatives of the individual functions, and then used the product rule to find the derivative of the product. The final derivative was simplified to 12x - 4.

Example Problems

  1. Find the derivative of the function f(x) = (2x + 5)(x - 3).
  2. Find the derivative of the function f(x) = (x + 2)(3x - 1).
  3. Find the derivative of the function f(x) = (x - 2)(2x + 5).

Step-by-Step Solutions

  1. Find the derivative of the function f(x) = (2x + 5)(x - 3).

First, let's find the derivatives of the individual functions.

ddx(2x+5)=2{ \frac{d}{dx}(2x + 5) = 2 }

ddx(x−3)=1{ \frac{d}{dx}(x - 3) = 1 }

Next, let's use the product rule to find the derivative of the product.

f′(x)=(2)(x−3)+(2x+5)(1){ f^{\prime}(x) = (2)(x - 3) + (2x + 5)(1) }

f′(x)=2x−6+2x+5{ f^{\prime}(x) = 2x - 6 + 2x + 5 }

f′(x)=4x−1{ f^{\prime}(x) = 4x - 1 }

  1. Find the derivative of the function f(x) = (x + 2)(3x - 1).

First, let's find the derivatives of the individual functions.

ddx(x+2)=1{ \frac{d}{dx}(x + 2) = 1 }

ddx(3x−1)=3{ \frac{d}{dx}(3x - 1) = 3 }

Next, let's use the product rule to find the derivative of the product.

f′(x)=(1)(3x−1)+(x+2)(3){ f^{\prime}(x) = (1)(3x - 1) + (x + 2)(3) }

f′(x)=3x−1+3x+6{ f^{\prime}(x) = 3x - 1 + 3x + 6 }

f′(x)=6x+5{ f^{\prime}(x) = 6x + 5 }

  1. Find the derivative of the function f(x) = (x - 2)(2x + 5).

First, let's find the derivatives of the individual functions.

ddx(x−2)=1{ \frac{d}{dx}(x - 2) = 1 }

ddx(2x+5)=2{ \frac{d}{dx}(2x + 5) = 2 }

Next, let's use the product rule to find the derivative of the product.

f′(x)=(1)(2x+5)+(x−2)(2){ f^{\prime}(x) = (1)(2x + 5) + (x - 2)(2) }

f′(x)=2x+5+2x−4{ f^{\prime}(x) = 2x + 5 + 2x - 4 }

f′(x)=4x+1{ f^{\prime}(x) = 4x + 1 }

Final Answer

Introduction

In our previous article, we explored how to find the derivative of a product of two functions using the product rule. In this article, we will answer some common questions related to finding derivatives of products of functions.

Q: What is the product rule?

A: The product rule is a fundamental concept in calculus that allows us to find the derivative of a product of two functions. The product rule states that if we have two functions, f(x) and g(x), then the derivative of their product is given by:

f′(x)g(x)+f(x)g′(x){ f^{\prime}(x)g(x) + f(x)g^{\prime}(x) }

Q: How do I apply the product rule to find the derivative of a product of two functions?

A: To apply the product rule, you need to follow these steps:

  1. Find the derivatives of the individual functions that make up the product.
  2. Use the product rule formula to find the derivative of the product.
  3. Simplify the derivative to get the final answer.

Q: What if I have a product of three or more functions? Can I still use the product rule?

A: Yes, you can still use the product rule to find the derivative of a product of three or more functions. However, you will need to apply the product rule multiple times to find the derivative of each individual function, and then combine the results using the product rule formula.

Q: Can I use the product rule to find the derivative of a product of functions that are not polynomials?

A: Yes, you can use the product rule to find the derivative of a product of functions that are not polynomials. However, you will need to use the chain rule and the product rule together to find the derivative of each individual function.

Q: What if I have a product of functions that involves a constant? Can I still use the product rule?

A: Yes, you can still use the product rule to find the derivative of a product of functions that involves a constant. However, you will need to remember that the derivative of a constant is zero.

Q: Can I use the product rule to find the derivative of a product of functions that involves a trigonometric function?

A: Yes, you can use the product rule to find the derivative of a product of functions that involves a trigonometric function. However, you will need to use the chain rule and the product rule together to find the derivative of each individual function.

Q: What if I have a product of functions that involves a logarithmic function? Can I still use the product rule?

A: Yes, you can still use the product rule to find the derivative of a product of functions that involves a logarithmic function. However, you will need to use the chain rule and the product rule together to find the derivative of each individual function.

Q: Can I use the product rule to find the derivative of a product of functions that involves an exponential function?

A: Yes, you can use the product rule to find the derivative of a product of functions that involves an exponential function. However, you will need to use the chain rule and the product rule together to find the derivative of each individual function.

Q: What if I have a product of functions that involves a rational function? Can I still use the product rule?

A: Yes, you can still use the product rule to find the derivative of a product of functions that involves a rational function. However, you will need to use the chain rule and the product rule together to find the derivative of each individual function.

Conclusion

In this article, we answered some common questions related to finding derivatives of products of functions. We hope that this article has been helpful in clarifying any confusion you may have had about the product rule and its application.

Example Problems

  1. Find the derivative of the function f(x) = (2x + 5)(x - 3).
  2. Find the derivative of the function f(x) = (x + 2)(3x - 1).
  3. Find the derivative of the function f(x) = (x - 2)(2x + 5).

Step-by-Step Solutions

  1. Find the derivative of the function f(x) = (2x + 5)(x - 3).

First, let's find the derivatives of the individual functions.

ddx(2x+5)=2{ \frac{d}{dx}(2x + 5) = 2 }

ddx(x−3)=1{ \frac{d}{dx}(x - 3) = 1 }

Next, let's use the product rule to find the derivative of the product.

f′(x)=(2)(x−3)+(2x+5)(1){ f^{\prime}(x) = (2)(x - 3) + (2x + 5)(1) }

f′(x)=2x−6+2x+5{ f^{\prime}(x) = 2x - 6 + 2x + 5 }

f′(x)=4x−1{ f^{\prime}(x) = 4x - 1 }

  1. Find the derivative of the function f(x) = (x + 2)(3x - 1).

First, let's find the derivatives of the individual functions.

ddx(x+2)=1{ \frac{d}{dx}(x + 2) = 1 }

ddx(3x−1)=3{ \frac{d}{dx}(3x - 1) = 3 }

Next, let's use the product rule to find the derivative of the product.

f′(x)=(1)(3x−1)+(x+2)(3){ f^{\prime}(x) = (1)(3x - 1) + (x + 2)(3) }

f′(x)=3x−1+3x+6{ f^{\prime}(x) = 3x - 1 + 3x + 6 }

f′(x)=6x+5{ f^{\prime}(x) = 6x + 5 }

  1. Find the derivative of the function f(x) = (x - 2)(2x + 5).

First, let's find the derivatives of the individual functions.

ddx(x−2)=1{ \frac{d}{dx}(x - 2) = 1 }

ddx(2x+5)=2{ \frac{d}{dx}(2x + 5) = 2 }

Next, let's use the product rule to find the derivative of the product.

f′(x)=(1)(2x+5)+(x−2)(2){ f^{\prime}(x) = (1)(2x + 5) + (x - 2)(2) }

f′(x)=2x+5+2x−4{ f^{\prime}(x) = 2x + 5 + 2x - 4 }

f′(x)=4x+1{ f^{\prime}(x) = 4x + 1 }

Final Answer

The final answer is: 12x−4\boxed{12x - 4}