Simple Differentiation Problem

In summary, the conversation discusses a differentiation problem and the different methods that can be used to solve it, including the product and chain rule. The conversation also mentions the use of a program called Maple that can assist with solving these types of problems.
  • #1
Vendoskt
5
0
I labeled this topic as a "Simple Differentiation Problem" because I know that it is simple, I'm just having problems with it.

The question is to differentiate ((x^2 + 2)^3)(x - 3)

The answer the book I'm using gives is ((x^2 + 2)^2)(7x^2 - 18x + 2)

Would this be differentiated by using a combination of product rule and chain rule? If that is the case then...

y' = ((x^2 + 2)^3) + (x - 3)(3(x^2 + 2)^2)(2x)

simplified a little

y' = ((x^2 + 2)^3) + 6x(x - 3)((x^2 + 2)^2) or

y' = ((x^2 + 2)^3) + (6x^2 - 18x)((x^2 + 2)^2)

this is where I am getting stuck. Either I am seriously lacking in my algebra skills, or I just used the wrong method to solve the problem.

Please help

Thank you
 
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  • #2
((x^2 + 2)^3) + (6x^2 - 18x)((x^2 + 2)^2)

you might try factoring a (x^2+2) out of ((x^2+2)^3)
 
  • #3
When in doubt you could expand the entire expression:
[tex] (x^2+2)^3(x-3) [/tex]

Instead think of it as two terms:
[tex] A\times B [/tex]
[tex] A = (x^2+2)^3 [/tex]
[tex] B = (x-3) [/tex]

Thus to differentiate [itex] \frac{d}{dx} [AB] = \frac{d}{dx}(A) B + A\frac{d}{dx}(B) [/itex]

So the same logic applies:
[tex] \frac{d}{dx}\left( (x^2+2)^3(x-3) \right) = \frac{d}{dx} \left[ (x^2+2)^3\right] (x-3) + \frac{d}{dx}\left[(x-3)\right](x^2+2)^3 [/tex]

To take care of that [itex] \frac{d}{dx} \left[(x^2+2)^3\right] [/itex] you use the chain rule.

So yup. You use the product and chain rule, or you could expand the entire thing out and do it that way... which might take awhile.

I don't know if you have ever used the program Maple. But it has a really nice tutor that walks you through differentiating. It's pretty cool, and quite helpful for these problems. It doesn't just spit out the correct answer, it instead shows you how to get it.
 

1. What is simple differentiation?

Simple differentiation is a mathematical process used in calculus to find the rate at which one quantity changes with respect to another. It is used to calculate the slope of a curve at a given point, and is an essential tool in many scientific and engineering applications.

2. How do you solve a simple differentiation problem?

To solve a simple differentiation problem, you need to follow a set of rules called the derivative rules, which outline the steps to take when differentiating different types of functions. These rules involve finding the derivative of each term in the function and combining them using algebraic operations.

3. What are the applications of simple differentiation?

Simple differentiation has many applications in different fields of science and engineering. It is used in physics to calculate velocities and accelerations, in biology to model population growth and decay, and in economics to analyze demand and supply curves, among others.

4. What is the difference between simple differentiation and partial differentiation?

The main difference between simple differentiation and partial differentiation is the number of variables involved. Simple differentiation is used to find the derivative of a function with one independent variable, while partial differentiation is used for functions with multiple independent variables. In simple differentiation, all other variables are treated as constants, while in partial differentiation, they are also differentiated.

5. Can simple differentiation be applied to non-linear functions?

Yes, simple differentiation can be applied to non-linear functions. The derivative of a non-linear function will result in another function that represents the slope of the original function at a given point. However, the derivative rules for non-linear functions can be more complex than those for linear functions, and may require the use of advanced techniques such as the chain rule or implicit differentiation.

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