MATH110 December 2013
• Q1 (a) • Q1 (b) • Q1 (c) • Q1 (d) • Q2 • Q3 (a) • Q3 (b) • Q4 (a) • Q4 (b) • Q4 (c) • Q5 (a) • Q5 (b) • Q6 (a) • Q6 (b) • Q6 (c) • Q7 • Q8 • Q9 • Q10 (a) • Q10 (b) • QS01 10(a) • QS01 10(b) •
Question 06 (b)

Consider the function: $f(x)={\dfrac {e^{\left(x^{3}\right)}e^{\left(2x^{2}\right)}e^{x}}{e^{3}}}.$
(b) Find the derivative of $f(x)$.

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 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

Simplify the function first as was done in part (a) and then use the chain rule.

Hint 2

You can also differentiate this using logarithmic differentiation.

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Solution 1

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Since we have simplified, we can simply apply the chain rule to it using $u=x^{3}+2x^{2}+x3$.
${\begin{aligned}f(u)&=e^{u}&f'(u)&=e^{u}\\u(x)&x^{3}+2x^{2}+x3&u'(x)&=3x^{2}+4x+1.\end{aligned}}$
Combining, we get:
$f'(x)=\left(3x^{2}+4x+1\right)\cdot e^{x^{3}+2x^{2}+x3}$

Solution 2

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Alternatively, we can start with the initial function and use logdiff:
${\begin{aligned}f(x)&={\frac {e^{\left(x^{3}\right)}e^{\left(2x^{2}\right)}e^{x}}{e^{3}}}\\\ln(f(x))&=\ln \left({\frac {e^{\left(x^{3}\right)}e^{\left(2x^{2}\right)}e^{x}}{e^{3}}}\right)\\&=\ln \left(e^{\left(x^{3}\right)}\right)+\ln \left(e^{\left(2x^{2}\right)}\right)+\ln \left(e^{x}\right)\ln \left({e^{3}}\right)\\&=x^{3}\ln \left(e\right)+2x^{2}\ln \left(e\right)+x\ln \left(e\right)3\ln \left(e\right)\\\ln(f(x))&=x^{3}+2x^{2}+x3\\{\frac {\rm {d}}{{\rm {d}}x}}\ln(f(x))&={\frac {\rm {d}}{{\rm {d}}x}}\left(x^{3}+2x^{2}+x3\right)\\{\frac {f'(x)}{f(x)}}&=3x^{2}+4x+1\\f'(x)&=\left(3x^{2}+4x+1\right)f(x)\\f'(x)&=\left(3x^{2}+4x+1\right)\cdot {\frac {e^{\left(x^{3}\right)}e^{\left(2x^{2}\right)}e^{x}}{e^{3}}}\\\end{aligned}}$

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MER QGH flag, MER QGQ flag, MER QGS flag, MER QGT flag, MER Tag Chain rule, MER Tag Logarithmic differentiation, Pages using DynamicPageList3 parser function, Pages using DynamicPageList3 parser tag

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