Find f'(x) Two Ways: Product/Quotient Rule & Simplifying

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In summary, we can find f'(x) two ways: by using the product or quotient rule, and by simplifying first. When simplifying first, we get the equation f'(x) = -27/x^4, which is equivalent to the answer given in the book.
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Homework Statement


Find f '(x) two ways: By using the product or quotient rule, and by simplifying first.
f(x) = ((x^3) + 9)/ (x^3)

Homework Equations


f '(x)= (G(x) * F '(x) - F(x) * G '(x)) / [G(x)^2]


The Attempt at a Solution


f '(x)= (x^3)(3x^2) - (x^3 + 9)(3x^2) / (x^3)^2 Plug into quotient rule
f '(x)= (3x^5) - (3x^5) - (27x^2) / (x^6) Simplify
f '(x)= (-27x^2)/(x^6) Canceled out 3x^5
f '(x)= (-27)/ (x^4) Simplified exponents

The answer in the back of the book is what I have(f '(x)= -27/(x^4) =-27x^-4)

So am I done? Was I supposed to simplify from the original equation for f(x) also? I guess I'm asking what "simplify first" exactly means. Any help is greatly appreciated, thank you!
 
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  • #2
simplify first probably means,

f(x) = (x^3 + 9)/x^3 = 1 + 9/x^3 = 1 + 9x^(-3),
so f'(x) = -27x^(-4)
 

Related to Find f'(x) Two Ways: Product/Quotient Rule & Simplifying

What is the product rule method for finding f'(x)?

The product rule method for finding f'(x) is used when the function is a product of two other functions. It states that the derivative of a product of two functions is equal to the first function multiplied by the derivative of the second function, plus the second function multiplied by the derivative of the first function.

What is the quotient rule method for finding f'(x)?

The quotient rule method for finding f'(x) is used when the function is a quotient of two other functions. It states that the derivative of a quotient of two functions is equal to the denominator multiplied by the derivative of the numerator, minus the numerator multiplied by the derivative of the denominator, all divided by the square of the denominator.

How do you simplify the derivative using the product rule?

To simplify the derivative using the product rule, you need to identify the two functions that make up the product, take the derivative of each function, and then plug those values into the product rule formula. Finally, you can combine like terms and simplify the resulting expression.

How do you simplify the derivative using the quotient rule?

To simplify the derivative using the quotient rule, you need to identify the two functions that make up the quotient, take the derivative of each function, and then plug those values into the quotient rule formula. Finally, you can combine like terms and simplify the resulting expression.

Why is it important to know both the product and quotient rule methods?

It is important to know both the product and quotient rule methods because not all functions can be easily differentiated using the basic rules of derivatives. By knowing these two methods, you can find the derivative of any function that is a product or a quotient of two other functions. This allows you to solve more complex problems and expand your understanding of calculus.

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