Science:Math Exam Resources/Courses/MATH110/December 2013/Question 03 (b)

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MATH110 December 2013

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Question 03 (b)
Rewrite $f(x)$ in such a way that you can differentiate it using a different rule than the Power Rule. Name the rule of differentiation which is applicable, and then confirm that the derivative calculated using this alternative method is the same as in part (a). (Bonus) You will receive one bonus mark for each different differentiation rule you use to differentiate $\displaystyle f(x)$.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
There are lots of choices here. The easier ones to see are the chain rule, the definition of a derivative, and using logarithmic differentiation. There are of course many others as you will see in the solution.

Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

Solution

Parts b and c combined: (These are all the rules I can think of). Addition rule: ${\begin{aligned}f(x)&={\frac {1}{2}}x^{6}+{\frac {1}{2}}x^{6}&f'(x)&=3x^{5}+3x^{5}\end{aligned}}$ Subtraction rule: ${\begin{aligned}f(x)&=2x^{6}-x^{6}&f'(x)&=12x^{5}-6x^{5}\end{aligned}}$ Product rule: ${\begin{aligned}f(x)&=x^{2}\cdot x^{4}&f'(x)&=2x\cdot x^{4}+4x^{3}\cdot x^{2}\end{aligned}}$ Quotient rule: ${\begin{aligned}f(x)&={\frac {x^{7}}{x}}&f'(x)&={\frac {7x^{6}\cdot x-x^{7}\cdot 1}{x^{2}}}\end{aligned}}$ Chain rule: ${\begin{aligned}f(x)&=(x^{3})^{2}&f'(x)&=3x^{2}\cdot 2(x^{3})\end{aligned}}$ Log-diff: ${\begin{aligned}\ln(f(x))&=6\ln(x)&f'(x)&={\frac {6}{x}}\cdot x^{6}\end{aligned}}$ Exp-chain rule: ${\begin{aligned}f(x)&=\exp[6\ln(x)]&f'(x)&={\frac {6}{x}}\cdot \exp[6\ln(x)]\end{aligned}}$ Limit definition of derivative: ${\begin{aligned}f'(x)&=\lim _{h\rightarrow 0}{\frac {(x+h)^{6}-x^{6}}{h}}&f'(x)&=\lim _{h\rightarrow 0}{\frac {6x^{5}h+15x^{4}h^{2}+\ldots +h^{6}}{h}}\\f'(x)&=\lim _{a\rightarrow x}{\frac {a^{6}-x^{6}}{a-x}}&f'(x)&=\lim _{a\rightarrow x}{\frac {(a-x)(a^{5}+a^{4}x+\ldots x^{5})}{(a-x)}}\end{aligned}}$

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