Chain Rule for Derivatives: Differentiating a Product with Chain Rule

In summary, the conversation discusses the method for differentiating the function f(x)=(3x^{2}+4)^{3}(5-3x)^{4}, which involves using the chain rule due to its product form. However, the product rule can also be used by expanding the function.
  • #1
mg0stisha
225
0

Homework Statement


Differentiate [tex] f(x)=(3x^{2}+4)^{3}(5-3x)^{4}[/tex]



Homework Equations


N/A



The Attempt at a Solution


I can see that this derivative is a product, yet also involves using chain rule. With this being said, am i just supposed to evaluate these separately using chain rule for each then multiply the results together? Or is there another way to differentiate this? Thanks in advance.
 
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  • #2
Take it as [tex]f(x) = g(u(x))h(v(x))[/tex]. Then [tex]f'(x) = g'(u)h(v) + h'(v)g(u)[/tex] where for example [tex] g'(u) = \frac{dg(u)}{dx} = \frac{dg(u)}{du}*\frac{du}{dx}[/tex], your standard chain rule.
 
  • #3
Ah yes I see it now, thank you very much!
 
  • #4
This is a product rule question, however, to take the derivative of this, you'll need the derivative of first and the derivative of the 2nd, thus the chain rule.

If you don't want to use the chain rule, you can expand both and use the product rule. :)
 

1. What is the chain rule in derivative calculus?

The chain rule is a formula used to find the derivative of a composite function. It states that the derivative of a composite function is equal to the derivative of the outer function, multiplied by the derivative of the inner function.

2. How do you apply the chain rule in finding derivatives?

To apply the chain rule, you must first identify the outer and inner functions in the composite function. Then, take the derivative of the outer function and multiply it by the derivative of the inner function. Finally, substitute the inner function into the derivative of the outer function.

3. What is the purpose of the chain rule in calculus?

The chain rule is an important tool in calculus as it allows us to find the derivative of complex functions that are composed of multiple simpler functions. It is essential in many applications, such as in physics, engineering, and economics.

4. Can the chain rule be applied to any composite function?

Yes, the chain rule can be applied to any composite function, as long as the inner and outer functions are differentiable. This means that they have a well-defined derivative at every point in their domain.

5. What are some common mistakes when using the chain rule?

One common mistake is not correctly identifying the inner and outer functions in the composite function. Another mistake is forgetting to multiply the derivative of the outer function by the derivative of the inner function. It is also crucial to be careful with chain rule notation, such as properly using the chain rule symbol and including parentheses where necessary.

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