# Differentiating $600x^{99}-55x^{54}+1$

Say we have the function

$$600x^{99}-55x^{54}+1$$

The answer to "differentiate" it is apparently

$$6x^{100}-x^{55}+x$$

but I'm not getting that answer.

A few questions:

1. How do you decide whether to use the basic, chain, or product rule when differentiating?

2. In the case of this particular question, which rule is necessary, and how do we apply it?

• you mean integrating, not differentiating. – Masacroso Mar 18 '17 at 21:37
• Do you mean "integrate?" – Pawel Mar 18 '17 at 21:37
• i'm doing a worksheet and it says "differentiate" !! i thought it was integration as well, and that would make sense, but i guess you have confirmed its a typo. @Paquarian – blue sky Mar 18 '17 at 21:40
• How do you know there is a typo?? – Alexey Mar 18 '17 at 21:59

## 3 Answers

Not sure if you want to integrate or differentiate but you are dealing with a polynomial in the form of $f(x)$ here.

I see from your comment that you are told on your worksheet to differentiate and if it was about integration, there would also be an arbitrary constant $C$ in the answer you've been given.

Anyway, if you take your problem the other way around, notice that you can use the basic differentiation rules that you know and try to differentiate:

$$6x^{100}-x^{55}+x$$

you should get:

$$600x^{99}-55x^{54}+1\;.$$

How do you decide whether to use the basic, chain, or product rule when differentiating?

• Use the (general) Power rule on a basic function $f(x)$.

• Use the Product rule whenever you have to differentiate a product of functions $f(x)g(x)$.

• Use the Quotien rule whenever you have to differentiate a ratio of functions $\frac{f(x)}{g(x)}$.

• Use the Chain rule whenever you have to differentiate a function inside a function $f(g(x))$.

Note that the Power, Product and Quotient rules are all special cases of the Chain rule.

As the other answers have said, there's probably a typo in the problem statement, since $$\frac{\mathrm d}{\mathrm dx}(6x^{100}-x^{55}+x)=600x^{99}−55x^{54}+1,$$ rather than the other way around. If you got that $$\frac{\mathrm d}{\mathrm dx}(600x^{99}−55x^{54}+1)=59400x^{98}-2970x^{53},$$ then you differentiated correctly. As for your other questions:

1) In general, when you get stuck on any sort of math problem it helps to divide things into manageable chunks. When it comes to differentiating a function, look for smaller functions inside that you already know how to differentiate, such as trig functions, logs, exponential functions, and things of the form $x^n$. Then use the rules you learned in calculus to combine them: \begin{align} (x^n)'&=nx^{n-1}\\ \big(k\cdot f(x)\big)'&=k\cdot f'(x)\\ \big(f(x)+g(x)\big)'&=f'(x)+g'(x)\\ \big(f(x)\cdot g(x)\big)'&=f'(x)\cdot g(x)+f(x)\cdot g'(x)\\ \big(f(g(x))\big)'&=f'(g(x))\cdot g'(x), \end{align} where $f$ and $g$ are functions, and $k$ and $n$ are constants.

2) As for this case specifically, there are three terms being added and you (should) know how to differentiate each individually. Thus there's no need to get fancy and use the product rule or chain rule. You simply use the power rule a few times, and add up what you get.

1) You use basic differentiation for basic situations. You use the product rule when you recognize products, and you use the chain rule for compositions. Thus, to differentiate $\sin x$ you use basic differentiation. To differentiate $\sin x \cos x$ you use the product rule, and to differentiate $\sin(\cos x)$ you use the chain rule.

2) In the case of your question, it is basic differentiation.