I am emailing a link of this to everyone on the class list every Friday afternoon. If you are not receiving these emails or want to have them sent to another email address feel free to email me at jpmccarthymaths@gmail.ie and I will add you to the mailing list.

## Notes

So far we have covered the quotient rule: MATH6015 notes.

## Answers to Exercises

### Page 13

Q. 1(a) False, (b) False ,(c) False. Q. 3 $12,\,16,\,3a^2-a+2$,$3a^2+a+2,\,3a^2+5a+4,\,6a^2-2a+4$,$12a^2-2a+2, \,3a^4-a^2+2,\,9a^4-6a^3+13a^2-4a+4$, $3h^2+6ah-h+3a^2-a+2$. Q. 4 $\displaystyle \frac{4}{3}\pi(1+3r+3r^2)$. Q. 5 $A(x)=x(10-x)$. Q. 7 (a) $y=2x+c$ with $c\in\mathbb{R}$, (b) $y=mx+(1-2m)$  for $m\in\mathbb{R}$, (c) $y=2x-3$. 8 (b) a change of $1^\circ$ C means a change of $\displaystyle\frac95$ F.

### Page 21

Q. 1( a) 59, (b) 256, (c) 0. Q.2 (a) $\displaystyle \frac35$, (b) $\displaystyle\frac65$, (c) $\displaystyle \frac32$, (d) 32.

### Page 30

(i) $2x-2$, (ii) $2x+5$, (iii) $3$, (iv) $4x-5$, (v) $2-2x$

### Page 32

Q. 1. (i) 5, (ii) $40x^7-10x^7$, (iii) $\displaystyle -\frac{7\sqrt{10}}{x^8}$, (iv) $\displaystyle -\frac{2}{5x^{7/5}}$ Q. 2 $10x^4$. Q. 3 $3-10x$ . Q. 4 $8x-24$. Q. 5 $18x+6$. Q. 6 $\displaystyle 3x^2+\frac{1}{\sqrt{x}}$. Q. 7 $\displaystyle \frac{3\sqrt{x}}{2}+\frac{1}{\sqrt{x}}$. Q. 8 $1029x^2+294x+21$. Q. 9 $\displaystyle \frac{2}{u^2}+2u+3u^2$. Q. 10 $\displaystyle y=-\frac14 x+1$.

### Page 34

Q. 1 $\cos x+10\sec^2x$. Q. 2 $y=x+1$.  Q. 3 $x=2$ and $-3$.

### Page 37

Q. 1 $x\cos x+\sin x$. Q. 2 $\displaystyle \frac{\cos x}{x^2}-\frac{2\sin x}{x^2}$. Q. 3 $\displaystyle \frac{1}{2\sqrt{x}}\sin x+\sqrt{x}\cos x$. Q. 4 $e^x(3-\sin x)+e^x(\cos x+3x)$. Q. 5 $\displaystyle \frac{1}{2\sqrt{x}}\log x+\frac{1}{\sqrt{x}}$. Q. 6  $\displaystyle \frac{1}{x^3}-\frac{2\log x}{x^3}$. Q. 7 $y=e$. Q. 8 $y=0$.

### Page 40

Q. 1 $\displaystyle \frac{5}{(2x+1)^2}$. Q. 2 $\displaystyle -\frac{t^6+3t^4+6t^2+2}{(t^4-2)^2}$. Q. 3 $\displaystyle -\frac{4x^3+2x}{(x^4+x^2+1)}$. Q. 4 $\displaystyle \frac{2t-2t^2}{(3t^2-2t+1)^2}$. Q. 5 $\displaystyle \frac{1}{\sqrt{x}(\sqrt{x}+1)^2}$. Q. 6 $\displaystyle \frac{m}{(1+mx)^2}$. Q. 7 $\displaystyle -\frac{x^2+1}{(x^2-1)^2}$. Q. 8 $\displaystyle \frac{x\cos x}{(x+\cos x)^2}$. Q. 9 $\displaystyle -\frac{4e^{2x}}{(e^{2x}-1)^2}$. Q. 10 $\displaystyle \frac{1+\log (2)}{u(1+\log(2u))^2}$. Q. 11 $\displaystyle \frac{1-2\log(x)}{x^3}$. Q. 12 $\displaystyle 0$. Q. 13 $\displaystyle y=\frac12 x+\frac12$. Q. 14 $\displaystyle y=-x+1$. Q. 15 $\displaystyle \frac{1}{2\sqrt{x}}-3$.

## Next Week

We will be doing the Chain Rule. This is very important for differentiation and we need to be good at the product rule and the quotient rule before we start it. Therefore we will have a tutorial on Monday. If we cover the Chain Rule with time to spare we may be as well to have another tutorial. We’ll see.

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