Calculus 1 Differentiation Problem: Chain Rule with Binomial Theorem Application

In summary, the conversation is about solving a differentiation problem involving the function f(x) = x(3x-9)^3. The correct approach is to use the product rule first, then the chain rule. However, there is also a more efficient method where one can skip the use of the product rule by rewriting the function as (3x^(4/3) - 9x^(1/3))^3 and applying the chain rule.
  • #1
Husaaved
19
1
I'm not entirely sure if this belongs in homework or elsewhere -- I'm self-teaching working through a basic calculus text, so it's not homework per se. In any case it's a simple differentiation problem wherein I am supposed to differentiate:

f(x) = x(3x-9)^3
f'(x) = 3x(3)(3x-9)^2 Applying chain rule
f'(x) = 9x(3x-9)^2

I know this isn't the correct answer.

I was half tempted to multiply out using the binomial theorem but I suspect there's a more efficient way to solve this. How am I to treat the x coefficient? Evidently not as a constant.
 
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  • #2
Did you see the product rule?

[tex](fg)^\prime = f^\prime g + fg^\prime[/tex]
 
  • #3
You have to use the product rule first, then the chain rule.

Of, since [itex]x= (x^{1/3})^3[/tex], f(x)= (x^{1/3}(3x- 9))^3= (3x^{4/3}- 9x^{1/3})^3
and now use the chain rule.
 
  • #4
HallsofIvy said:
You have to use the product rule first, then the chain rule.

Or, you could skip the need for the product rule by looking at it this way ...

Since [itex]x= (x^{1/3})^3[/itex],
then [itex]f(x)= (x^{1/3}(3x- 9))^3= (3x^{4/3}- 9x^{1/3})^3[/itex]
and now use the chain rule.
There, itex fixed. :wink:
 
  • #5
Thanks!:redface:
 

1. What is the chain rule in Calculus 1?

The chain rule is a fundamental concept in calculus that allows us to find the derivative of a function composed of two or more nested functions. It is used to find the rate of change of one variable with respect to another variable.

2. Why is the chain rule important in Calculus 1?

The chain rule is important because it allows us to find the derivative of complex functions, which are often encountered in real-world applications. It is also essential in understanding the relationship between different variables and how they affect each other.

3. How do you use the chain rule in Calculus 1?

To use the chain rule, you need to identify the outer function and the inner function in a given composite function. Then, you take the derivative of the outer function and multiply it by the derivative of the inner function. This process is repeated for multiple nested functions.

4. What are some common mistakes when using the chain rule in Calculus 1?

One common mistake is not properly identifying the outer and inner functions. Another mistake is not correctly applying the chain rule formula, such as forgetting to multiply by the derivative of the inner function. It is also essential to simplify the expression before taking the derivative to avoid any errors.

5. How can I practice and improve my understanding of the chain rule in Calculus 1?

You can practice by solving a variety of problems that involve the chain rule, such as finding derivatives of composite functions, using the chain rule to find the slope of a tangent line, and applying it to real-world scenarios. Additionally, seeking help from a tutor or studying from reliable resources can also enhance your understanding of the chain rule.

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