Free Printable CHAP 9: DIFFERENTIATION (KMS)

1.

Given $y=3e^{x^4},\$ then $\frac{\text{d}y}{\text{d}x}=p\times3e^{x^4}$ .What is $p?$

a)

$x^4$

b)

$4x^3$

c)

$12x^3$

d)

$12x^4$

2.

Given $y=2e^{x^2+1}.$ Then, $\frac{d}{dx}\left(2e^{x^2+1}\right)=p\times2e^{x^2+1}$ . Which fits the role of $p?$

a)

$4x$

b)

$4$

c)

$2$

d)

$2x$

3.

Given $y=e^{\sqrt[]{x}},$ then y' is $pe^{\sqrt[]{x}}$ . What is $p?$

a)

$\sqrt[]{x}$

b)

$2\sqrt[]{x}$

c)

$\frac{1}{2}\sqrt[]{x}$

d)

$\frac{1}{2\sqrt[]{x}}$

4.

$y=2^{1-3x}$ . Then, $\frac{\text{d}y}{\text{d}x}=2^{1-3x}\ \ln a\ \times b$ .

What are the CORRECT values of $a,b?$

a)

$a=2,b=3$

b)

$a=3,b=2$

c)

$a=2,b=-3$

d)

$a=-3,b=2$

5.

Given $y=2e^{3x}-7^x$ . Then, $\frac{\text{d}y}{\text{d}x}=pe^{3x}-7^x\ln q.$

What satisfies $p,q?$

a)

$p=2,q=x$

b)

$p=3,q=7$

c)

$p=6,q=x$

d)

$p=6,q=7$

6.

Given $y=2xe^{3x}.$ Then, $\frac{\text{d}y}{\text{d}x}=pe^{3x}+2x\left(q\right).$

Which BEST represents $p,q?$

a)

$p=2,q=e^{3x}$

b)

$p=3,q=e^{3x}$

c)

$p=2,q=3e^{3x}$

d)

$p=3,q=3e^{3x}$

7.

$\frac{\text{d}}{\text{d}x}\left(\frac{x-1}{e^{2x}}\right)=\frac{e^{2x}\left(1\right)-\left(x-1\right)\left(e^{2x}\right)}{\left(e^{2x}\right)^2}$ .

It is believed there is AN ERROR in the above working. If yes, which term?

a)

$e^{2x}\left(1\right)$

b)

$\left(x-1\right)e^{2x}$

c)

$\left(e^{2x}\right)^2$

d)

Impossible, all is OK

8.

Given $y=\left(1-e^{2x}\right)^3,$ then $\frac{\text{d}y}{\text{d}x}=3\left(1-e^{2x}\right)^2\left(-e^{2x}\right).$

There is A MISTAKE in one of the terms. What is the MISTAKE?

a)

$3$

b)

$\left(1-e^{2x}\right)$

c)

$\left(1-e^{2x}\right)^2$

d)

$-e^{2x}$

9.

Given $y=7^{3x},$ then $\frac{\text{d}y}{\text{d}x}=7^{3x}\left(\ln\ a\right)b.$

What is $a?$

a)

$3$

b)

$3x$

c)

$7$

d)

$7^{3x}$

10.

Given $y=7^{3x},$ then $\frac{\text{d}y}{\text{d}x}=7^{3x}\left(\ln\ a\right)b.$

What is $b?$

a)

$3$

b)

$3x$

c)

$7$

d)

$7^{3x}$

11.

Given $y=5^{x^2}.$ Then, $\frac{\text{d}y}{\text{d}x}=5^{x^2}\ln5\ \frac{\text{d}}{\text{d}x}\left(q\right)$ .

What is $q?$

a)

$5$

b)

$x^2$

c)

$1$

d)

$2x$

12.

Given $y=\ln\left(2x\right),$

$\frac{\text{d}y}{\text{d}x}=?$

a)

$\frac{1}{2x}$

b)

$\frac{1}{x}$

c)

$\ln2x$

d)

$\ln2$

13.

$y=2\sin x-3\cot x$

$\frac{d}{dx}\left(y\right)=\frac{d}{dx}\left(2\sin\ x\right)-\frac{d}{dx}\left(3\ \cot\ x\right)$

$p=q+r$

What is $p?$

a)

$1$

b)

$y$

c)

$\frac{\text{d}y}{\text{d}x}$

d)

$\frac{\text{d}x}{\text{d}y}$

14.

$y=2\sin x-3\cot x$

$\frac{d}{dx}\left(y\right)=\frac{d}{dx}\left(2\sin\ x\right)-\frac{d}{dx}\left(3\ \cot\ x\right)$

$p=q+r$

What is $q?$

a)

$2\sin\ x$

b)

$-2\cos\ x$

c)

$\text{cos x}$

d)

$\text{}2\cos x$

15.

$y=2\sin x-3\cot x$

$\frac{d}{dx}\left(y\right)=\frac{d}{dx}\left(2\sin\ x\right)-\frac{d}{dx}\left(3\ \cot\ x\right)$

$p=q+r$

What is $r?$

a)

$-3\operatorname{cosec}^2x$

b)

$\operatorname{cosec}^2x$

c)

$3\operatorname{cosec}^2x$

d)

$-\operatorname{cosec}^2x$

16.

Given $y=\cos\ 4x.$ Then, $y'=4\sin\ 4x$ .

It was realized the value of $4$ is INCORRECT. What is the CORRECT value?

a)

$1$

b)

$-1$

c)

$-4$

d)

Disagree.It is indeed correct

17.

If $y=-4\tan x-3\sec4x,$ then

$\frac{d}{dx}\left(-4\tan x-3\sec4x\right)=a\sec^2x+b\sec4x\tan4x$ .

What is $a,b?$

a)

$a=4,b=3$

b)

$a=-4,b=-3$

c)

$a=-4,b=12$

d)

$a=-4,b=-12$

18.

Given $y=\cot\ 7x,\ \frac{dy}{dx}=a\ \operatorname{cosec}^2\ 7x$ .

What is $a?$

a)

$1$

b)

$-1$

c)

$7$

d)

$-7$

19.

Given $y=2\cos^34x.\$ Then, $y=2\left(\cos4x\right)^3$ .

Given $\frac{\text{d}y}{\text{d}x}=6\left(\cos4x\right)^2\left(\sin4x\right)$ .

Which is INCORRECT term?

a)

$6$

b)

$\cos4x$

c)

$\left(\cos4x\right)^2$

d)

$\left(\sin4x\right)$

20.

Given $y=\sin^24x$ .

Then, $\frac{\text{d}y}{\text{d}x}=2\left(\sin4x\right)q$ .

What is $q?$

a)

$\cos4x$

b)

$-\cos4x$

c)

$4\cos4x$

d)

$-4\cos4x$

21.

Given $y=3\tan^22x.$

Then, $\frac{\text{d}y}{\text{d}x}=6\left(\tan2x\right)q$ .What is $q?$

a)

$\sec^22x$

b)

$-\sec^22x$

c)

$2\sec^22x$

d)

$-2\sec^22x$

22.

$\frac{d}{dx}\left(\frac{\sin x}{\cos x}\right)=\frac{\left(\cos x\right)p-\sin x\left(q\right)}{r}$ .

What is $p?$

a)

$\sin x$

b)

$-\sin x$

c)

$\cos\ x$

d)

$-\cos x$

23.

$\frac{d}{dx}\left(\frac{\sin x}{\cos x}\right)=\frac{\left(\cos x\right)p-\sin x\left(q\right)}{r}$ .

What is $q?$

a)

$\sin x$

b)

$-\sin x$

c)

$\cos\ x$

d)

$-\cos x$

24.

$\frac{d}{dx}\left(\frac{\sin x}{\cos x}\right)=\frac{\left(\cos x\right)p-\sin x\left(q\right)}{r^2}$ .

What is $r?$

a)

$\sin x$

b)

$-\sin x$

c)

$\cos\ x$

d)

$-\cos x$

25.

Given $y=\ln\left(\left(\sqrt[]{x^3+1}\right)\left(1-2x^2\right)\right).$

Which shows the CORRECT use of law of log to ensure READY to find $\frac{\text{d}y}{\text{d}x}$ ?

a)

$y=\ln\sqrt[]{x^3+1}+\ln\left(1-2x^2\right)^3$

b)

$y=\ln\sqrt[]{x^3+1}+3\ln\left(1-2x^2\right)^{ }$

c)

$y=2\ln\left(x^3+1\right)+3\ln\left(1-2x^2\right)^{ }$

d)

$y=\frac{1}{2}\ln\left(x^3+1\right)+3\ln\left(1-2x^2\right)^{ }$

26.

$y=\frac{1}{2}\ln\left(x^3+1\right)+3\ln\left(1-2x^2\right)$

$\frac{dy}{dx}=\frac{1}{2}\left(\frac{a}{b}\right)+3\left(\frac{c}{d}\right)$

What is $a?$

a)

$1$

b)

$3x^2$

c)

$x^3$

d)

$x^3+1$

27.

$y=\frac{1}{2}\ln\left(x^3+1\right)+3\ln\left(1-2x^2\right)$

$\frac{dy}{dx}=\frac{1}{2}\left(\frac{a}{b}\right)+3\left(\frac{c}{d}\right)$

What is $b?$

a)

$1$

b)

$3x^2$

c)

$x^3$

d)

$x^3+1$

28.

$y=\frac{1}{2}\ln\left(x^3+1\right)+3\ln\left(1-2x^2\right)$

$\frac{dy}{dx}=\frac{1}{2}\left(\frac{a}{b}\right)+3\left(\frac{c}{d}\right)$

What is $c?$

a)

$1$

b)

$4x$

c)

$-4x$

d)

$1-2x^2$

29.

$y=\frac{1}{2}\ln\left(x^3+1\right)+3\ln\left(1-2x^2\right)$

$\frac{dy}{dx}=\frac{1}{2}\left(\frac{a}{b}\right)+3\left(\frac{c}{d}\right)$

What is $d?$

a)

$1$

b)

$4x$

c)

$-4x$

d)

$1-2x^2$

30.

Given $y=3x\ln2x.$

Then, $\frac{dy}{dx}=3\ln2x+b$ .

What is $b?$

a)

$\frac{3}{2}$

b)

$3$

c)

$0$

d)

$\frac{3}{x}$

31.

Given $y=\ln\left(\frac{x^2\left(1-2x\right)}{2x+1}\right).$

Choose the form which is READY for differentiation

a)

$y=\ln x^2\left(1-2x\right)-\ln\left(2x+1\right)$

b)

$y=\ln x^2+\ln\left(1-2x\right)-\ln\left(2x+1\right)$

c)

$y=2\ln x+\ln\left(1-2x\right)-\ln\left(2x+1\right)$

d)

$y=2\ln x+\ln\left(1-2x\right)+\ln\left(2x+1\right)$

32.

Given $y=\ln\left(\frac{\sqrt[]{x+2}}{\sqrt[3]{x-3}}\right)$ .

Choose the form which is READY for differentiation.

a)

$y=2\ln\left(x+2\right)-3\ln\left(x-3\right)$

b)

$y=\ln\sqrt[]{x+2}-\ln\sqrt[3]{x-3}$

c)

$y=\frac{1}{2}\ln\left(x+2\right)-\frac{1}{3}\ln\left(x-3\right)$

d)

$y=\frac{1}{2}\ln\left(x+2\right)-\frac{1}{2}\ln\left(x-3\right)$

33.

$\frac{d}{dx}\left(\ln\left(1-2x\right)\right)=\frac{a}{1-2x}$ .

What is the value of $a?$

a)

$1$

b)

$2$

c)

$-1$

d)

$-2$

34.

$\ln y=\ln x$

$\frac{d}{dx}\left(\ln y\right)=\frac{d}{dx}\left(\ln x\right)$

$p=r$ .

What is $p?$

a)

$1$

b)

$\frac{1}{y}$

c)

$\frac{1}{y}\left(\frac{dy}{dx}\right)$

d)

$\frac{dy}{dx}$

35.

$\ln y=\ln x$

$\frac{d}{dx}\left(\ln y\right)=\frac{d}{dx}\left(\ln x\right)$

$p=r$ .

What is $r?$

a)

$1$

b)

$\frac{1}{x}$

c)

$\ln\ x$

d)

$\frac{1}{\ln x}$

36.

Given $\frac{d}{dx}\left(3\ln x-2\ln\left(2-3x\right)\right)=\frac{3}{x}-\frac{6}{2-3x}$ .

There is a term which is INCORRECT. Which one?

a)

$3$

b)

$x$

c)

$-6$

d)

$2-3x$

37.

$\frac{d}{dx}\left(x^3+y^3\right)=\frac{d}{dx}\left(\sin\ x\ +\cos\ y\right)$

$3x^2+3y^2p=\cos x+q$

Which represents $p,q$ CORRECTLY?

a)

$p=1,q=-\sin y\frac{\text{d}y}{\text{d}x}$

b)

$p=\frac{dy}{dx},q=\sin y$

c)

$p=1,q=\sin y$

d)

$p=\frac{dy}{dx},q=-\sin y\frac{\text{d}y}{\text{d}x}$

38.

Choose the BEST method to perform differentiation for below.

$y=\sin2x$

a)

Product Rule

b)

Quotient rule

c)

Power rule

d)

Chain rule

39.

Choose the BEST method to perform differentiation for below.

$y=\frac{1}{3}\sin2x$

a)

Product Rule

b)

Quotient rule

c)

Power rule

d)

Chain rule

40.

Choose the BEST method to perform differentiation for below.

$y=\frac{x}{3}\sin x$

a)

Product Rule

b)

Quotient rule

c)

Power rule

d)

Chain rule

41.

Choose the BEST method to perform differentiation for below.

$y=\sin^3x$

a)

Product Rule

b)

Quotient rule

c)

Power rule

d)

Chain rule

42.

Choose the BEST method to perform differentiation for below.

$y=\sin^{ }x\cos x$

a)

Product Rule

b)

Quotient rule

c)

Power rule

d)

Chain rule

43.

Choose the BEST method to perform differentiation for below.

$y=\sin^{ }x\left(\cos x\right)$

a)

Product Rule

b)

Quotient rule

c)

Power rule

d)

Chain rule

44.

Given

$xy+2y=2$ .

What is the technique used to differentiate term $xy?$

a)

Product rule

b)

Quotient rule

c)

Direct

d)

Chain rule

45.

Given

$\left(\sin2y+2\right)^2+3y=x^2$ .

What is the rule to differentiate term $\left(\sin2y+2\right)^2?$

a)

Power and direct

b)

Power and Chain

c)

Power and Power

d)

Direct and Direct

46.

Given $e^{xy}+\sin y=3$ .

What is the rule to differentiate term $e^{xy}?$

a)

Direct exponential formula

b)

Direct exponential formula, product rule

c)

Direct exponential formula,chain rule

d)

Direct exponential formula, direct rule

47.

Given $3\sec^3x-y\cos y+3x=1$ .

What is the BEST rule to differentiate term $3\sec^3x\ ?$

a)

Direct rule

b)

Power rule

c)

Product rule

d)

Quotient rule

48.

Given $3\sec^3x-y\cos y+3x=1$ .

What is the BEST rule to differentiate term $y\cos y\ ?$

a)

Direct rule

b)

Power rule

c)

Product rule

d)

Quotient rule

49.

Given $\frac{d}{dx}\left(y^2\right)=2yq$ .

What is $q?$

a)

$\frac{dy}{dx}$

b)

$1$

c)

$y^2$

d)

-

50.

$y\ln x=e^{x-y}$

$p\ln x+y\left(q\right)=\frac{d}{dx}\left(x-y\right)e^{x-y}$

$p\ln x+\frac{y}{x}=\left(1-r\right)e^{x-y}$

What is $p?$

a)

$1$

b)

$\frac{dy}{dx}$

c)

$y^2$

d)

$y$

51.

$y\ln x=e^{x-y}$

$p\ln x+y\left(q\right)=\frac{d}{dx}\left(x-y\right)e^{x-y}$

$p\ln x+\frac{y}{x}=\left(1-r\right)e^{x-y}$

What is $q?$

a)

$\ln x$

b)

$x$

c)

$\frac{1}{x}$

d)

$1$

52.

$y\ln x=e^{x-y}$

$p\ln x+y\left(q\right)=\frac{d}{dx}\left(x-y\right)e^{x-y}$

$p\ln x+\frac{y}{x}=\left(1-r\right)e^{x-y}$

What is $r?$

a)

$y$

b)

$1$

c)

$x$

d)

$\frac{dy}{dx}$

53.

$\frac{1}{y}-\frac{1}{x}=3$

$y^p-x^{-1}=3$

$-y^{-2}q+x^{-2}=0$

What is $p?$

a)

$-1$

b)

$-2$

c)

$0$

d)

$1$

54.

$\frac{1}{y}-\frac{1}{x}=3$

$y^p-x^{-1}=3$

$-y^{-2}q+x^{-2}=0$

What is $q?$

a)

$1$

b)

$x^{-1}$

c)

$\frac{1}{x^2}$

d)

$\frac{dy}{dx}$

55.

$\sin\ y=y\cos\ x$

$a\frac{dy}{dx}=yb+c\ \cos x$

What is $a?$

a)

$\sin y$

b)

$-\sin y$

c)

$\cos y$

d)

$-\cos y$

56.

$\sin\ y=y\cos\ x$

$a\frac{dy}{dx}=yb+c\ \cos x$

What is $b?$

a)

$\cos x$

b)

$-\cos x$

c)

$\sin x$

d)

$-\sin x$

57.

$\sin\ y=y\cos\ x$

$a\frac{dy}{dx}=yb+c\ \cos x$

What is $c?$

a)

$1$

b)

$\frac{dy}{dx}$

c)

$y^2$

d)

$-1$

58.

$\cos\left(x+y\right)=2x\sin\ y$

$-\left(1+\frac{dy}{dx}\right)t=2\sin y+\cos y\ \frac{dy}{dx}$ .

Based on above process, $t=?$

a)

$\cos\left(x+y\right)$

b)

$\sin\left(x+y\right)$

c)

$-\cos\left(x+y\right)$

d)

$-\sin\left(x+y\right)$

59.

Given $y=\tan\left(2x\right),$

what is $\frac{\text{d}y}{\text{d}x}=?$

a)

$\sec^2\left(2x\right)$

b)

$2\sec^2\left(2x\right)$

c)

$2x\ \sec^2\left(2x\right)$

d)

$-2\ \sec^2\left(2x\right)$

60.

$x=\tan y$

$\frac{d}{dx}\left(x\right)=\frac{d}{dx}\left(\tan y\right)$

$p=q$

What is $p?$

a)

$1$

b)

$\frac{dy}{dx}$

c)

$\frac{dx}{dy}$

d)

$0$

61.

$x=\tan y$

$\frac{d}{dx}\left(x\right)=\frac{d}{dx}\left(\tan y\right)$

$p=q$

What is $q?$

a)

$\tan y$

b)

$\tan y\ \frac{dy}{dx}$

c)

$\sec^2y\ \ \frac{dy}{dx}$

d)

$\sec^2y$

62.

Given

$x=\tan y$

$\frac{d}{dy}\left(x\right)=\frac{d}{dy}\left(\tan y\right)$

$p=q$

What is $p?$

a)

$1$

b)

$\frac{dy}{dx}$

c)

$\frac{dx}{dy}$

d)

$0$

63.

Given

$x=\tan y$

$\frac{d}{dy}\left(x\right)=\frac{d}{dy}\left(\tan y\right)$

$p=q$

What is $q?$

a)

$\tan\ y$

b)

$\tan\ y\ \frac{dy}{dx}$

c)

$\sec^2y\ \frac{dy}{dx}$

d)

$\sec^2y$

CHAP 9: DIFFERENTIATION (KMS)

Total questions: 63

Worksheet time: 1hrs 3mins

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