Limits and Derivatives
Miscellaneous Exercise Solutions
Differentiation Rules:
1. First Principle: $f'(x) = \lim_{h \to 0} \frac{f(x+h) – f(x)}{h}$
2. Product Rule: $(uv)’ = u’v + uv’$
3. Quotient Rule: $(\frac{u}{v})’ = \frac{u’v – uv’}{v^2}$
4. Power Rule: $\frac{d}{dx}(ax+b)^n = n(ax+b)^{n-1} \cdot a$
1. First Principle: $f'(x) = \lim_{h \to 0} \frac{f(x+h) – f(x)}{h}$
2. Product Rule: $(uv)’ = u’v + uv’$
3. Quotient Rule: $(\frac{u}{v})’ = \frac{u’v – uv’}{v^2}$
4. Power Rule: $\frac{d}{dx}(ax+b)^n = n(ax+b)^{n-1} \cdot a$
Q1
1. Find derivatives from first principle:
(i) $f(x) = -x$
Limit
$\lim_{h \to 0} \frac{-(x+h) – (-x)}{h} = \lim_{h \to 0} \frac{-x-h+x}{h} = \lim_{h \to 0} \frac{-h}{h} = -1$.
(ii) $f(x) = (-x)^{-1} = -\frac{1}{x}$
Setup
$f(x+h) = -\frac{1}{x+h}$.
Limit
$\lim_{h \to 0} \frac{1}{h} [-\frac{1}{x+h} – (-\frac{1}{x})] = \lim_{h \to 0} \frac{1}{h} [\frac{1}{x} – \frac{1}{x+h}]$.
Simplify
$\lim_{h \to 0} \frac{x – (x+h)}{h x(x+h)} = \lim_{h \to 0} \frac{-h}{h x(x+h)} = \frac{-1}{x^2}$.
(iii) $f(x) = \sin(x+1)$
Limit
$\lim_{h \to 0} \frac{\sin(x+h+1) – \sin(x+1)}{h}$. Use $\sin C – \sin D = 2\cos\frac{C+D}{2}\sin\frac{C-D}{2}$.
Expand
$\lim_{h \to 0} \frac{2\cos(x+1+h/2)\sin(h/2)}{h}$.
Result
$\cos(x+1) \cdot \lim_{h \to 0} \frac{\sin(h/2)}{h/2} = \cos(x+1)$.
(iv) $f(x) = \cos(x – \frac{\pi}{8})$
Limit
$\lim_{h \to 0} \frac{\cos(x+h-\pi/8) – \cos(x-\pi/8)}{h}$. Use $\cos C – \cos D = -2\sin\frac{C+D}{2}\sin\frac{C-D}{2}$.
Expand
$\lim_{h \to 0} \frac{-2\sin(x-\pi/8+h/2)\sin(h/2)}{h} = -\sin(x-\pi/8)$.
Q2-Q6
Find the derivative of the following functions:
2. $x+a$
$\frac{d}{dx}(x) + \frac{d}{dx}(a) = 1 + 0 = 1$.
3. $(px+q)(\frac{r}{x}+s)$
Expand first: $pr + psx + \frac{qr}{x} + qs$.
Deriv: $0 + ps – \frac{qr}{x^2} + 0 = ps – \frac{qr}{x^2}$.
Deriv: $0 + ps – \frac{qr}{x^2} + 0 = ps – \frac{qr}{x^2}$.
4. $(ax+b)(cx+d)^2$
Product Rule: $(ax+b)'(cx+d)^2 + (ax+b)[(cx+d)^2]’$
$= a(cx+d)^2 + (ax+b) \cdot 2c(cx+d) = (cx+d)[a(cx+d) + 2c(ax+b)]$.
$= (cx+d)(3acx + ad + 2bc)$.
$= a(cx+d)^2 + (ax+b) \cdot 2c(cx+d) = (cx+d)[a(cx+d) + 2c(ax+b)]$.
$= (cx+d)(3acx + ad + 2bc)$.
5. $\frac{ax+b}{cx+d}$
Quotient Rule: $\frac{a(cx+d) – c(ax+b)}{(cx+d)^2} = \frac{acx + ad – acx – bc}{(cx+d)^2} = \frac{ad-bc}{(cx+d)^2}$.
6. $\frac{1 + \frac{1}{x}}{1 – \frac{1}{x}}$
Simplify to $\frac{x+1}{x-1}$.
Quotient Rule: $\frac{(1)(x-1) – (1)(x+1)}{(x-1)^2} = \frac{x-1-x-1}{(x-1)^2} = \frac{-2}{(x-1)^2}$.
Quotient Rule: $\frac{(1)(x-1) – (1)(x+1)}{(x-1)^2} = \frac{x-1-x-1}{(x-1)^2} = \frac{-2}{(x-1)^2}$.
Q7-Q10
Find the derivative of the following functions:
7. $\frac{1}{ax^2+bx+c}$
Chain Rule on $(ax^2+bx+c)^{-1}$: $-(ax^2+bx+c)^{-2} \cdot (2ax+b) = \frac{-(2ax+b)}{(ax^2+bx+c)^2}$.
8. $\frac{ax+b}{px^2+qx+r}$
$\frac{a(px^2+qx+r) – (ax+b)(2px+q)}{(px^2+qx+r)^2}$.
Num: $apx^2+aqx+ar – (2apx^2+aqx+2bpx+bq) = -apx^2 – 2bpx + ar – bq$.
Num: $apx^2+aqx+ar – (2apx^2+aqx+2bpx+bq) = -apx^2 – 2bpx + ar – bq$.
9. $\frac{px^2+qx+r}{ax+b}$
$\frac{(2px+q)(ax+b) – a(px^2+qx+r)}{(ax+b)^2}$.
Num: $2apx^2+2bpx+aqx+bq – apx^2-aqx-ar = apx^2 + 2bpx + bq – ar$.
Num: $2apx^2+2bpx+aqx+bq – apx^2-aqx-ar = apx^2 + 2bpx + bq – ar$.
10. $\frac{a}{x^4} – \frac{b}{x^2} + \cos x$
Rewrite: $ax^{-4} – bx^{-2} + \cos x$.
Deriv: $-4ax^{-5} – (-2)bx^{-3} – \sin x = \frac{-4a}{x^5} + \frac{2b}{x^3} – \sin x$.
Deriv: $-4ax^{-5} – (-2)bx^{-3} – \sin x = \frac{-4a}{x^5} + \frac{2b}{x^3} – \sin x$.
Q11-Q15
Find the derivative of the following functions:
11. $4\sqrt{x} – 2$
$4x^{1/2} – 2 \Rightarrow 4(\frac{1}{2})x^{-1/2} = \frac{2}{\sqrt{x}}$.
12. $(ax+b)^n$
Chain/Power Rule: $n(ax+b)^{n-1} \cdot \frac{d}{dx}(ax+b) = na(ax+b)^{n-1}$.
13. $(ax+b)^n (cx+d)^m$
Product Rule:
$(ax+b)^n \cdot m(cx+d)^{m-1}c + (cx+d)^m \cdot n(ax+b)^{n-1}a$.
$(ax+b)^n \cdot m(cx+d)^{m-1}c + (cx+d)^m \cdot n(ax+b)^{n-1}a$.
14. $\sin(x+a)$
$\cos(x+a) \cdot \frac{d}{dx}(x+a) = \cos(x+a)$.
15. $\csc x \cot x$
Product Rule: $\csc x(-\csc^2 x) + \cot x(-\csc x \cot x)$
$= -\csc^3 x – \csc x \cot^2 x$.
$= -\csc^3 x – \csc x \cot^2 x$.
Limits and Derivatives
Miscellaneous Exercise Q16-Q30
Differentiation Rules Review:
Product Rule: $(uv)’ = u’v + uv’$
Quotient Rule: $(\frac{u}{v})’ = \frac{u’v – uv’}{v^2}$
Chain Rule Logic: $\frac{d}{dx}(\sin^n x) = n\sin^{n-1} x \cdot \cos x$
Product Rule: $(uv)’ = u’v + uv’$
Quotient Rule: $(\frac{u}{v})’ = \frac{u’v – uv’}{v^2}$
Chain Rule Logic: $\frac{d}{dx}(\sin^n x) = n\sin^{n-1} x \cdot \cos x$
Quotient Rule (Q16-Q21)
16. Find derivative of $\frac{\cos x}{1+\sin x}$
17. Find derivative of $\frac{\sin x + \cos x}{\sin x – \cos x}$
18. Find derivative of $\frac{\sec x – 1}{\sec x + 1}$
19. Find derivative of $\sin^n x$
20. Find derivative of $\frac{a+b\sin x}{c+d\cos x}$
21. Find derivative of $\frac{\sin(x+a)}{\cos x}$
Numerator
$u’v – uv’ = (-\sin x)(1+\sin x) – (\cos x)(\cos x)$.
Simplify
$-\sin x – \sin^2 x – \cos^2 x = -\sin x – 1 = -(1+\sin x)$.
Result
$\frac{-(1+\sin x)}{(1+\sin x)^2} = \frac{-1}{1+\sin x}$.
17. Find derivative of $\frac{\sin x + \cos x}{\sin x – \cos x}$
Numerator
$(\cos x – \sin x)(\sin x – \cos x) – (\sin x + \cos x)(\cos x + \sin x)$.
Expand
$-(\sin x – \cos x)^2 – (\sin x + \cos x)^2 = -(1 – \sin 2x) – (1 + \sin 2x) = -2$.
$\frac{-2}{(\sin x – \cos x)^2}$
18. Find derivative of $\frac{\sec x – 1}{\sec x + 1}$
Simplify
Convert to cos: $y = \frac{1-\cos x}{1+\cos x}$.
Quotient
Num: $(\sin x)(1+\cos x) – (1-\cos x)(-\sin x) = 2\sin x$.
$\frac{2\sin x}{(1+\cos x)^2}$
19. Find derivative of $\sin^n x$
Chain Rule
Treat as $(\sin x)^n$. Derivative is $n(\sin x)^{n-1} \cdot \frac{d}{dx}(\sin x)$.
$n \sin^{n-1} x \cos x$
20. Find derivative of $\frac{a+b\sin x}{c+d\cos x}$
u’, v’
$u’ = b\cos x, \quad v’ = -d\sin x$.
Num
$(b\cos x)(c+d\cos x) – (a+b\sin x)(-d\sin x)$.
Simplify
$bc\cos x + bd\cos^2 x + ad\sin x + bd\sin^2 x = bc\cos x + ad\sin x + bd$.
$\frac{bc\cos x + ad\sin x + bd}{(c+d\cos x)^2}$
21. Find derivative of $\frac{\sin(x+a)}{\cos x}$
Num
$\cos(x+a)\cos x – \sin(x+a)(-\sin x) = \cos(x+a)\cos x + \sin(x+a)\sin x$.
Identity
$\cos(A-B) \Rightarrow \cos(x+a-x) = \cos a$.
$\frac{\cos a}{\cos^2 x}$
Product Rule (Q22-Q25)
22. Find derivative of $x^4(5\sin x – 3\cos x)$
23. Find derivative of $(x^2+1)\cos x$
24. Find derivative of $(ax^2 + \sin x)(p + q\cos x)$
25. Find derivative of $(x + \cos x)(x – \tan x)$
Differentiate
$4x^3(5\sin x – 3\cos x) + x^4(5\cos x + 3\sin x)$.
$x^3[20\sin x – 12\cos x + 5x\cos x + 3x\sin x]$
23. Find derivative of $(x^2+1)\cos x$
Apply Rule
$(2x)(\cos x) + (x^2+1)(-\sin x)$.
$2x\cos x – (x^2+1)\sin x$
24. Find derivative of $(ax^2 + \sin x)(p + q\cos x)$
Apply Rule
$(2ax + \cos x)(p + q\cos x) + (ax^2 + \sin x)(-q\sin x)$.
$(2ax + \cos x)(p + q\cos x) – q\sin x(ax^2 + \sin x)$
25. Find derivative of $(x + \cos x)(x – \tan x)$
Apply Rule
$(1 – \sin x)(x – \tan x) + (x + \cos x)(1 – \sec^2 x)$.
$(1 – \sin x)(x – \tan x) – (x + \cos x)\tan^2 x$
Mixed Problems (Q26-Q30)
26. Find derivative of $\frac{4x + 5\sin x}{3x + 7\cos x}$
27. Find derivative of $\frac{x^2 \cos(\frac{\pi}{4})}{\sin x}$
28. Find derivative of $\frac{x}{1+\tan x}$
29. Find derivative of $(x+\sec x)(x-\tan x)$
30. Find derivative of $\frac{x}{\sin^n x}$
Num
$(4+5\cos x)(3x+7\cos x) – (4x+5\sin x)(3-7\sin x)$.
Expand
$12x + 28\cos x + 15x\cos x + 35\cos^2 x – (12x – 28x\sin x + 15\sin x – 35\sin^2 x)$.
$\frac{35 + 15x\cos x + 28\cos x + 28x\sin x – 15\sin x}{(3x + 7\cos x)^2}$
27. Find derivative of $\frac{x^2 \cos(\frac{\pi}{4})}{\sin x}$
Note
$\cos(\frac{\pi}{4}) = \frac{1}{\sqrt{2}}$ is a constant. Focus on $\frac{x^2}{\sin x}$.
Quotient
$\frac{2x\sin x – x^2\cos x}{\sin^2 x}$.
$\frac{1}{\sqrt{2}} \cdot \frac{x(2\sin x – x\cos x)}{\sin^2 x}$
28. Find derivative of $\frac{x}{1+\tan x}$
Num
$1(1+\tan x) – x(\sec^2 x)$.
$\frac{1+\tan x – x\sec^2 x}{(1+\tan x)^2}$
29. Find derivative of $(x+\sec x)(x-\tan x)$
Product
$(1+\sec x\tan x)(x-\tan x) + (x+\sec x)(1-\sec^2 x)$.
$(1+\sec x\tan x)(x-\tan x) – (x+\sec x)\tan^2 x$
30. Find derivative of $\frac{x}{\sin^n x}$
Num
$1(\sin^n x) – x(n\sin^{n-1} x \cos x)$.
Factor
$\sin^{n-1} x (\sin x – nx\cos x)$.
Simplify
$\frac{\sin^{n-1} x (\sin x – nx\cos x)}{(\sin^n x)^2} = \frac{\sin x – nx\cos x}{\sin^{n+1} x}$.
$\frac{\sin x – nx\cos x}{\sin^{n+1} x}$