# Solving Derivative Problem using Chain/Product Rule

• Apost8
In summary, the conversation discussed finding the derivative of a complicated function using the product and chain rules. The final answer involved simplifying algebraically. The person thanked the expert for reviewing their work and mentioned that they will need to get used to using LaTeX.
Apost8
I have a tricky derivative HW problem I'm working and am hoping someone might tell me if I'm doing this correctly or not. Thanks in advance!

Find g'(x) where g(x) = [(1+4x)^5] X [(-x^2+x+3)^8)]

By the product rule, I get:

g'x = [(d/dx((1+4x)^5)) X (-x^2+x+3)^8] + [(d/dx(-x^2+x+3)^8) X (1+4x)^5)]

Then, using the chain rule I get:

g'(x) = [((d/dx((1+4x)^5))(d/dx(1+4x))) X (-x^2+x+3)^8] + [((d/dx((-x^2+x+3)^8))(d/dx(-x^2+x+3))) X ((1+4x)^5)]

Giving:

g'(x) = [(20(1+4x)^4) X (-x^2+x+3)^8] + [(8(-x^2+x+3)^7 X (-2x+1)) X ((1+4x)^5)]

Then it should just be a matter of simplifying algebraically. Right?

Last edited:
That's correct (even if it's a bit tough on the eyes :) )

Thanks for looking over that mess for me. I'll have to get used to using LaTeX I suppose. :)

## What is the chain rule?

The chain rule is a mathematical rule used in calculus to find the derivative of composite functions. It states that for a function composed of two or more functions, the derivative is equal to the derivative of the outer function multiplied by the derivative of the inner function.

## What is the product rule?

The product rule is a formula used to find the derivative of a product of two functions. It states that the derivative of a product 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.

## How do I apply the chain rule to solve derivative problems?

To apply the chain rule, you need to 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. This will give you the overall derivative of the composite function.

## How do I use the product rule to find the derivative of a product?

To use the product rule, you need to identify the two functions that are being multiplied together. Then, take the derivative of each function and apply the formula: (first function)*(derivative of second function) + (second function)*(derivative of first function). This will give you the derivative of the product.

## Can I use both the chain rule and product rule at the same time?

Yes, you can use both the chain rule and product rule in the same problem if the function is composite and involves multiplication of two or more functions. You would first apply the chain rule to find the derivative of the composite function, and then use the product rule to find the derivative of the product function within the composite function.

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