Science:Math Exam Resources/Courses/MATH104/December 2014/Question 05 (c)

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MATH104 December 2014

  •  Q1 (a)  •  Q1 (b)  •  Q1 (c)  •  Q1 (d)  •  Q1 (e)  •  Q1 (f)  •  Q1 (g)  •  Q1 (h)  •  Q1 (i)  •  Q1 (j)  •  Q1 (k)  •  Q1 (l)  •  Q1 (m)  •  Q1 (n)  •  Q2  •  Q3 (a)  •  Q3 (b)  •  Q3 (c)  •  Q4  •  Q5 (a)  •  Q5 (b)  •  Q5 (c)  •  Q5 (d)  •  Q5 (e)  •  Q5 (f)  •  Q6  •  

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Question 05 (c)
Let $f(x)={\frac {e^{x}}{x^{2}}}$. Find $f''(x)$.

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
Take derivative of first derivative as found in part $a$.

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Solution

Recall from part $a$ that $f'(x)={\frac {e^{x}(x-2)}{x^{3}}}$. We take the derivative of $f'$ with respect to $x$ using the quotient rule. Let ${\begin{aligned}g(x)&=e^{x}(x-2)\quad \quad \quad g'(x)=e^{x}(x-2)+e^{x}=e^{x}(x-1)\\h(x)&=x^{3}\quad \quad \quad \quad \quad \quad h'(x)=3x^{2}\end{aligned}}$ By the quotient rule: ${\begin{aligned}f''(x)&={\frac {g'(x)h(x)-g(x)h'(x)}{(h(x))^{2}}}\\&={\frac {e^{x}(x-1)x^{3}-3e^{x}(x-2)x^{2}}{x^{6}}}\\&={\frac {e^{x}x^{2}(x(x-1)-3(x-2))}{x^{6}}}\\&={\frac {e^{x}(x^{2}-4x+6)}{x^{4}}}\end{aligned}}$

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