How Do You Apply the Chain and Product Rules in Differentiation?

In summary, the student attempted to solve a homework problem by using the chain rule and product rule, but was unable to simplify the equation.
  • #1
andrew.c
46
0

Homework Statement


Differentiate...
[tex](3x+4)^7 (7x-1)^3[/tex]
and
simplify

Homework Equations



Chain rule and Product rule

The Attempt at a Solution



I got (splitting the components up to substitute into the product rule) and using the chain rule

[tex]
\begin{align*}
\\f(x) = (3x+4)^7\\
f'(x) = 21(3x+4)^6\\
g(x) = (7x-1)^3\\
g'(x) = 21(7x-1)^2
\end{align*}
[/tex]

and so, using the product rule...
[tex]
\begin{align*}
\\f'(x)g(x) + f(x)g'(x)\\
=21(3x+4)^6 (7x-1)^3 + 21(3x+4)^7 (7x-1)^2\\
=(3x+4)^6 (7x-1)^3 + (3x+4)^7 (7x-1)^2\\
\end{align*}
[/tex]

and now I don't know how to simplify further.
I got it down to 10x+3, but this doesn't match the answer in the marking

[tex]21(3x+4)^6 (7x-1)^2 (10x+3)[/tex]

Any ideas guys?
 
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  • #2
andrew.c said:

Homework Statement


Differentiate...
[tex](3x+4)^7 (7x-1)^3[/tex]
and
simplify


Homework Equations



Chain rule and Product rule

The Attempt at a Solution



I got (splitting the components up to substitute into the product rule) and using the chain rule

[tex]
\begin{align*}
\\f(x) = (3x+4)^7\\
f'(x) = 21(3x+4)^6\\
g(x) = (7x-1)^3\\
g'(x) = 21(7x-1)^2
\end{align*}
[/tex]

and so, using the product rule...
[tex]
\begin{align*}
\\f'(x)g(x) + f(x)g'(x)\\
=21(3x+4)^6 (7x-1)^3 + 21(3x+4)^7 (7x-1)^2\\
=(3x+4)^6 (7x-1)^3 + (3x+4)^7 (7x-1)^2\\
\end{align*}
[/tex]

and now I don't know how to simplify further.
I got it down to 10x+3, but this doesn't match the answer in the marking

[tex]21(3x+4)^6 (7x-1)^2 (10x+3)[/tex]

Any ideas guys?
How could you possibly do this differentiation correctly (which you did) and not be able to multiply polynomials! You product, after multiplying out, will involve x9. It certainly is not "10x+ 3"!

Notice that your [itex](3x+4)^6(7x-1)^3+ (3x+4)^7(7x-1)^2[/itex] has at least 6 factors of 3x-4 and 2 factors of 7x-1 in each term. Take them out and you have left exactly that "10x+3" you mentioned.
 
  • #3
HallsofIvy said:
How could you possibly do this differentiation correctly (which you did) and not be able to multiply polynomials!

Yeah, just had a look through this again and that was a really stupid mistake! I guess that's what hours of maths can do to you!

Thanks
 

FAQ: How Do You Apply the Chain and Product Rules in Differentiation?

What is the purpose of differentiating and simplifying in science?

Differentiating and simplifying is a mathematical process used in science to make complex equations or relationships easier to understand and work with. It allows scientists to better analyze and interpret data, and make predictions based on their findings.

How do you differentiate and simplify an equation?

To differentiate an equation, you need to find its derivative by using mathematical rules and formulas. Once you have the derivative, you can simplify the equation by simplifying any constants or coefficients, and combining like-terms. This will result in a simpler and more manageable equation.

Can you give an example of differentiating and simplifying in science?

One example of differentiating and simplifying in science is in the field of physics, when calculating the acceleration of an object. The initial equation, F=ma (force = mass x acceleration), can be differentiated to find the object's velocity at any given time and then simplified to find its average velocity over a period of time.

What are some common mistakes when differentiating and simplifying in science?

One common mistake is forgetting to apply the chain rule when differentiating composite functions. Another mistake is not simplifying the equation enough, resulting in a more complex and confusing equation. It is also important to be careful with algebraic manipulations, as simple errors can lead to incorrect results.

Is it always necessary to differentiate and simplify in science?

No, it is not always necessary to differentiate and simplify in science. It depends on the specific problem or situation at hand. In some cases, it may be more beneficial to keep the equation in its original form, especially if the simplified version would lose important information or become too vague. However, in most cases, differentiating and simplifying can make equations easier to work with and understand.

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