Find The Derivative Of The Function Below:${ Y = 7x^2 E^{3x} }$A. ${ 21x^2 E^{3x} + 14x E^{3x} }$B. ${ 14x^2 E^{3x} + 21x E^{3x} }$C. ${ 28x^2 E^{3x} + 42x E^{3x} }$D. ${ 42x^2 E^{3x} + 28x E^{3x} }$

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Introduction

In calculus, the derivative of a function represents the rate of change of the function with respect to one of its variables. In this article, we will focus on finding the derivative of a given function, which is a combination of a polynomial and an exponential function. We will use the product rule and the chain rule to find the derivative of the function.

The Product Rule

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

ddx(f(x)g(x))=f′(x)g(x)+f(x)g′(x){ \frac{d}{dx} (f(x)g(x)) = f'(x)g(x) + f(x)g'(x) }

The Chain Rule

The chain rule is another important rule in calculus that allows us to find the derivative of a composite function. If we have a function, f(x), and we want to find the derivative of a composite function, f(g(x)), then the derivative is given by:

ddxf(g(x))=f′(g(x))⋅g′(x){ \frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x) }

The Given Function

The given function is:

y=7x2e3x{ y = 7x^2 e^{3x} }

This function is a combination of a polynomial and an exponential function. To find the derivative of this function, we will use the product rule and the chain rule.

Finding the Derivative

To find the derivative of the given function, we will first identify the two functions that we need to differentiate. In this case, the two functions are:

f(x)=7x2{ f(x) = 7x^2 } g(x)=e3x{ g(x) = e^{3x} }

Now, we will find the derivatives of these two functions:

f′(x)=14x{ f'(x) = 14x } g′(x)=3e3x{ g'(x) = 3e^{3x} }

Next, we will use the product rule to find the derivative of the given function:

ddx(7x2e3x)=f′(x)g(x)+f(x)g′(x){ \frac{d}{dx} (7x^2 e^{3x}) = f'(x)g(x) + f(x)g'(x) }

Substituting the values of f(x), g(x), f'(x), and g'(x), we get:

ddx(7x2e3x)=14xe3x+7x2â‹…3e3x{ \frac{d}{dx} (7x^2 e^{3x}) = 14x e^{3x} + 7x^2 \cdot 3e^{3x} }

Simplifying the expression, we get:

ddx(7x2e3x)=14xe3x+21x2e3x{ \frac{d}{dx} (7x^2 e^{3x}) = 14x e^{3x} + 21x^2 e^{3x} }

Conclusion

In this article, we found the derivative of a given function using the product rule and the chain rule. The derivative of the function is:

ddx(7x2e3x)=14xe3x+21x2e3x{ \frac{d}{dx} (7x^2 e^{3x}) = 14x e^{3x} + 21x^2 e^{3x} }

This is the correct answer among the options provided.

Answer

The correct answer is:

14x2e3x+21xe3x{ 14x^2 e^{3x} + 21x e^{3x} }

This is option B.

Final Thoughts

Introduction

In our previous article, we discussed how to find the derivative of a function using the product rule and the chain rule. In this article, we will provide a Q&A section to help clarify any doubts that readers may have.

Q: What is the derivative of a function?

A: The derivative of a function represents the rate of change of the function with respect to one of its variables. It is a measure of how fast the function changes as the variable changes.

Q: What is the product rule?

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

ddx(f(x)g(x))=f′(x)g(x)+f(x)g′(x){ \frac{d}{dx} (f(x)g(x)) = f'(x)g(x) + f(x)g'(x) }

Q: What is the chain rule?

A: The chain rule is another important rule in calculus that allows us to find the derivative of a composite function. If we have a function, f(x), and we want to find the derivative of a composite function, f(g(x)), then the derivative is given by:

ddxf(g(x))=f′(g(x))⋅g′(x){ \frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x) }

Q: How do I find the derivative of a function?

A: To find the derivative of a function, you need to identify the two functions that you need to differentiate. Then, you need to find the derivatives of these two functions using the product rule and the chain rule.

Q: What is the difference between the product rule and the chain rule?

A: The product rule is used to find the derivative of a product of two functions, while the chain rule is used to find the derivative of a composite function.

Q: Can you provide an example of how to use the product rule and the chain rule?

A: Let's say we have a function:

y=7x2e3x{ y = 7x^2 e^{3x} }

To find the derivative of this function, we need to use the product rule and the chain rule. First, we need to identify the two functions that we need to differentiate:

f(x)=7x2{ f(x) = 7x^2 } g(x)=e3x{ g(x) = e^{3x} }

Next, we need to find the derivatives of these two functions:

f′(x)=14x{ f'(x) = 14x } g′(x)=3e3x{ g'(x) = 3e^{3x} }

Now, we can use the product rule to find the derivative of the given function:

ddx(7x2e3x)=f′(x)g(x)+f(x)g′(x){ \frac{d}{dx} (7x^2 e^{3x}) = f'(x)g(x) + f(x)g'(x) }

Substituting the values of f(x), g(x), f'(x), and g'(x), we get:

ddx(7x2e3x)=14xe3x+7x2â‹…3e3x{ \frac{d}{dx} (7x^2 e^{3x}) = 14x e^{3x} + 7x^2 \cdot 3e^{3x} }

Simplifying the expression, we get:

ddx(7x2e3x)=14xe3x+21x2e3x{ \frac{d}{dx} (7x^2 e^{3x}) = 14x e^{3x} + 21x^2 e^{3x} }

Q: What is the final answer?

A: The final answer is:

14x2e3x+21xe3x{ 14x^2 e^{3x} + 21x e^{3x} }

This is option B.

Conclusion

In this article, we provided a Q&A section to help clarify any doubts that readers may have about finding the derivative of a function. We hope that this article has provided a clear and concise explanation of how to find the derivative of a function.