Science:Math Exam Resources/Courses/MATH100/December 2013/Question 06

MATH100 December 2013

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Question 06
Full-Solution Problem. Justify your answer and show your work. Full simplification of numerical answers is required unless explicitly stated otherwise. Using the definition of the derivative, compute $\displaystyle f'(x)$ if $\displaystyle f(x)={\frac {x}{2x+1}}$. No marks will be given for the use of differential rules, but you may use them to check your answer.

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 hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint.

Hint 1
The derivative of $f$ at $x$ is the slope of the line tangent to $f$ at $x$. What is the expression for the slope of a line?

Hint 2
The slope of a line between two points $\displaystyle (x_{1},f(x_{1}))$ and $\displaystyle (x_{2},f(x_{2}))$ is ${\frac {f(x_{2})-f(x_{1})}{x_{2}-x_{1}}}$. What happens as $\displaystyle x_{2}-x_{1}\to 0$? How can we express this mathematically?

Hint 3
The limit definition of the derivative of $f$ at $x$ is $f'(x)=\lim _{h\to 0}{\frac {f(x+h)-f(x)}{h}}$. Equivalently, the derivative of $f$ at $a$ is $f'(a)=\lim _{x\to a}{\frac {f(x)-f(a)}{x-a}}$.

Checking a solution serves two purposes: helping you if, after having used all the hints, 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 1

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. By the limit definition of the derivative of $f$ at $x$, ${\begin{aligned}f'(x)&=\lim _{h\to 0}{\frac {f(x+h)-f(x)}{h}}\\&=\lim _{h\to 0}{\frac {{\frac {x+h}{2(x+h)+1}}-{\frac {x}{2x+1}}}{h}}\\&=\lim _{h\to 0}{\frac {(x+h)(2x+1)-x(2(x+h)+1)}{h(2(x+h)+1)(2x+1)}}\\&=\lim _{h\to 0}{\frac {(2x^{2}+x+2xh+h)-(2x^{2}+2xh+x)}{h(2(x+h)+1)(2x+1)}}\\&=\lim _{h\to 0}{\frac {h}{h(2(x+h)+1)(2x+1)}}\\&=\lim _{h\to 0}{\frac {1}{(2(x+h)+1)(2x+1)}}\\&={\frac {1}{(2(x+0)+1)(2x+1)}}\\&={\color {blue}{\frac {1}{(2x+1)^{2}}}}\end{aligned}}$ Use of the quotient rule confirms that this is indeed $\displaystyle f'(x)$: ${\begin{aligned}f'(x)&={\frac {1\cdot (2x+1)-x\cdot 2}{(2x+1)^{2}}}\\&={\color {OliveGreen}{\frac {1}{(2x+1)^{2}}}}\end{aligned}}$

Solution 2

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. By the alternative limit definition of the derivative of $f$ at $a$, ${\begin{aligned}f'(a)&=\lim _{x\to a}{\frac {f(x)-f(a)}{x-a}}\\&=\lim _{x\to a}{\frac {{\frac {x}{2x+1}}-{\frac {a}{2a+1}}}{x-a}}\\&=\lim _{x\to a}{\frac {\frac {x(2a+1)-a(2x+1)}{(2x+1)(2a+1)}}{x-a}}\\&=\lim _{x\to a}{\frac {\frac {(2ax+x)-(2ax+a)}{(2x+1)(2a+1)}}{x-a}}\\&=\lim _{x\to a}{\frac {\frac {x-a}{(2x+1)(2a+1)}}{x-a}}\\&=\lim _{x\to a}{\frac {1}{(2x+1)(2a+1)}}\\&={\frac {1}{(2a+1)^{2}}}\\f'(x)&={\color {blue}{\frac {1}{(2x+1)^{2}}}}\end{aligned}}$ Use of the quotient rule confirms that this is indeed $\displaystyle f'(x)$: ${\begin{aligned}f'(x)&={\frac {1\cdot (2x+1)-x\cdot 2}{(2x+1)^{2}}}\\&={\color {OliveGreen}{\frac {1}{(2x+1)^{2}}}}\end{aligned}}$

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