ECAT Mathematics
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ECAT Mathematics
Chapter 16: Differentiation
75 Questions
90 Minutes
Pass: 60%+
ECAT Prep
0
The derivative f'(x) = lim_{h→0} [f(x+h)−f(x)]/h is:
A
Integral
B
Derivative (first principles)
C
Area under curve
D
Limit at infinity
1
d/dx(x^n) =
A
nx^{n+1}
B
nx^{n-1}
C
x^n lnx
D
n/x
2
d/dx(c) where c is constant =
A
c
B
0
C
cx
D
1
3
d/dx(sinx) =
A
−cosx
B
cosx
C
sinx
D
−sinx
4
d/dx(cosx) =
A
sinx
B
−sinx
C
cosx
D
−cosx
5
d/dx(tanx) =
A
sec²x
B
csc²x
C
secx tanx
D
−csc²x
6
d/dx(e^x) =
A
xe^{x-1}
B
e^x
C
e^x lne
D
0
7
d/dx(lnx) =
A
1/x
B
x
C
lnx
D
e^x
8
Product rule: d/dx[f(x)g(x)] =
A
f'g+fg'
B
f'g'
C
fg
D
f/g+g/f
9
Quotient rule: d/dx[f/g] =
A
(f'g+fg')/g²
B
(f'g−fg')/g²
C
f'g−fg'
D
f'g/g²
10
Chain rule: d/dx[f(g(x))] =
A
f'(x)×g'(x)
B
f'(g(x))×g'(x)
C
f(g'(x))
D
f'(x)+g'(x)
11
d/dx(x³+2x²−5x+1) =
A
3x²+4x−5
B
3x²+2x−5
C
x³+4x−5
D
3x+4
12
d/dx(sin(x²)) using chain rule =
A
cosx²
B
2x cos(x²)
C
2cos(x²)
D
−2x sin(x²)
13
d/dx(e^{x²}) =
A
e^{x²}
B
2x e^{x²}
C
x e^{x²}
D
2e^{x²}
14
d/dx(ln(x²)) =
A
2x
B
1/x²
C
2/x
D
2lnx
15
The second derivative f''(x) =
A
d/dx[f(x)]
B
d/dx[f'(x)]
C
∫f'(x)dx
D
f'(x)²
16
d/dx(x³) =
A
3x
B
3x²
C
x²
D
3
17
d/dx(√x) =
A
2√x
B
1/(2√x)
C
−1/(2x^{3/2})
D
x^{-1/2}
18
d/dx(1/x) =
A
1
B
−1/x²
C
1/x²
D
lnx
19
d/dx(sec x) =
A
secx tanx
B
−cscx cotx
C
sec²x
D
tanx
20
d/dx(cscx) =
A
cscx cotx
B
−cscx cotx
C
csc²x
D
−csc²x
21
d/dx(cotx) =
A
csc²x
B
−csc²x
C
sec²x
D
−sec²x
22
d/dx(a^x) =
A
xa^{x-1}
B
a^x ln a
C
a^x
D
a^x/lna
23
d/dx(log_a x) =
A
1/(x ln a)
B
lnx/a
C
1/(a lnx)
D
a/x
24
d/dx(sin⁻¹x) =
A
1/√(1−x²)
B
-1/√(1−x²)
C
1/√(1+x²)
D
1/(1+x²)
25
d/dx(cos⁻¹x) =
A
1/√(1−x²)
B
−1/√(1−x²)
C
1/√(1+x²)
D
1/(1+x²)
26
d/dx(tan⁻¹x) =
A
1/(1+x²)
B
−1/(1+x²)
C
1/√(1−x²)
D
1/(1−x²)
27
If f(x)=x² sinx: f'(x) =
A
2x sinx+x² cosx
B
2x sinx−x² cosx
C
2x cosx
D
x² cosx+sinx
28
If f(x)=sinx/x: f'(x) =
A
(cosx×x−sinx)/x²
B
(cosx−sinx)/x
C
cosx/x²
D
sinx/x²
29
Implicit differentiation of x²+y²=r²: dy/dx =
A
−x/y
B
x/y
C
y/x
D
−y/x
30
Tangent to curve y=x² at x=2: slope =
A
2
B
4
C
8
D
1
31
If y=x³−3x, stationary points where y'=0: 3x²−3=0→x=
A
x=0
B
x=±1
C
x=1 only
D
x=3
32
At x=1: y''=6x=6>0: the point is a:
A
Maximum
B
Minimum
C
Inflection
D
Saddle
33
At x=−1: y''=6(−1)=−6<0: the point is a:
A
Maximum
B
Minimum
C
Inflection
D
Saddle
34
The chain rule applied to y=f(u) where u=g(x): dy/dx =
A
dy/du × du/dx
B
dy/dx × du/dx
C
dy/du + du/dx
D
dy/du / du/dx
35
d/dx(e^{sinx}) =
A
e^{sinx} cosx
B
cosx e^{cosx}
C
e^{cosx}
D
e^{sinx}
36
d/dx(ln(cosx)) =
A
−tanx
B
tanx
C
−sinx/cosx=−tanx
D
Both A and C
37
d²y/dx² for y=x⁴: first y'=4x³, then y''=
A
4x³
B
12x²
C
12x
D
4x⁴
38
If y=x^n, the nth derivative is:
A
n!
B
n!x
C
n! x^0=n!
D
nx^{n-1}
39
Critical points occur where:
A
f'(x)=0 only
B
f'(x)=0 or f'(x) undefined
C
f(x)=0
D
f''(x)=0
40
Point of inflection occurs where:
A
f'(x)=0
B
f'(x) changes sign
C
f''(x) changes sign
D
f(x)=0
41
Rolle's Theorem: if f is continuous on [a,b], differentiable on (a,b), and f(a)=f(b), then ∃c∈(a,b) with:
A
f(c)=0
B
f'(c)=0
C
f'(c)=f(a)
D
f(c)=f(a)
42
Mean Value Theorem: f'(c) =
A
f(b)/f(a)
B
[f(b)−f(a)]/(b−a)
C
f(a)/b
D
f'(a)+f'(b)
43
d/dx(x sin x + cos x) =
A
x cosx
B
x cosx+sinx−sinx=x cosx
C
sinx+x cosx−sinx
D
Both A and B (=x cosx)
44
For y = e^{ax}: dy/dx =
A
e^{ax}
B
ae^{ax}
C
a e^{ax}
D
e^{ax}/a
45
Parametric: x=t², y=t³. dy/dx =
A
dy/dt ÷ dx/dt = 3t²/(2t)=3t/2
B
2t/3t²=2/(3t)
C
3t/(2)
D
t³/t²=t
46
If f(x)=x²+3x, the instantaneous rate of change at x=2:
A
7
B
4
C
5
D
10
47
d/dx(x e^x) =
A
e^x
B
xe^x+e^x
C
xe^x
D
e^x(x+1)
48
d/dx[ln(x²+1)] =
A
1/(x²+1)
B
2x/(x²+1)
C
x/(x²+1)
D
2/(x²+1)
49
The slope of normal to y=x² at x=3 is:
A
6
B
−1/6
C
1/6
D
−6
50
Higher-order derivative: if f(x)=sinx, f'''(x) =
A
sinx
B
−sinx
C
cosx
D
−cosx
51
d/dx(tan(3x²)) =
A
sec²(3x²)
B
6x sec²(3x²)
C
3x² sec²(3x²)
D
sec²(3x)×6x
52
Increasing function: f'(x) ? 0
A
<
B
=
C
>
D
≤
53
Decreasing function: f'(x) ? 0
A
<
B
=
C
>
D
≤
54
Second derivative test for maxima: f'(c)=0 and f''(c):
A
=0
B
<0
C
>0
D
undefined
55
Second derivative test for minima: f'(c)=0 and f''(c):
A
=0
B
<0
C
>0
D
undefined
56
d/dx(x⁴−4x³+6x²−4x+1) = (x−1)^4 diff:
A
4(x−1)³
B
4x³−12x²+12x−4
C
Both A and B
D
4x³
57
The velocity v = dx/dt. Acceleration =
A
dx/dt
B
d²x/dt²
C
dv/dt
D
Both B and C
58
If position s=5t²−3t+2, velocity at t=2:
A
20−3=17
B
10−3=7
C
10+3=13
D
5(4)−3=17
59
d/dx(√(1+x²)) =
A
x/√(1+x²)
B
1/(2√(1+x²))
C
x√(1+x²)
D
1+x²
60
d/dx(x/sinx) using quotient rule:
A
(sinx−x cosx)/sin²x
B
(cosx−sinx)/x²
C
x cosx/sin²x
D
1/cosx
61
The gradient of a curve at a point is the:
A
y-intercept
B
slope of the tangent
C
x-intercept
D
area under curve
62
For minimum material box optimization (calculus): the condition is:
A
f(x)=0
B
f'(x)=0
C
f''(x)=0
D
f'(x)>0
63
d/dx(x^x) = ? (using logarithmic differentiation)
A
x^x
B
x^x(1+lnx)
C
x^x lnx
D
x^{x-1}
64
d/dx(logₑ(sinx)) =
A
cosx/sinx=cotx
B
sinx/cosx
C
1/sinx
D
−cotx
65
d/dx(cos²x) =
A
−2cosx sinx
B
2cosx sinx
C
−sin²x
D
−2sinx cosx = −sin(2x)
66
The derivative of f(x)=|x| at x=0:
A
0
B
1
C
−1
D
Does not exist
67
L'Hopital applied to lim_{x→0} x/sinx: d/dx(x)=1, d/dx(sinx)=cosx. Limit=
A
1/cos0=1
B
0/1=0
C
cos0/1=1
D
Both A and C
68
d/dx(sin²x+cos²x) =
A
0
B
2sinx cosx
C
−2sinx cosx
D
1
69
For implicit differentiation of xy=1: y+x(dy/dx)=0→dy/dx=
A
y/x
B
−y/x
C
x/y
D
−x/y
70
For y=1/x: dy/dx=−y/x=−(1/x)/x=−1/x². Verify:
A
Correct (power rule gives −1/x²)
B
Incorrect
C
Only for x>0
D
Only at x=1
71
d/dx[f(x)]^n =
A
n[f(x)]^{n-1}
B
n[f(x)]^{n-1} f'(x)
C
[f(x)]^{n-1} f'(x)
D
n[f(x)]^n f'(x)
72
d/dx(sin(cosx)) =
A
cos(cosx)×sinx
B
−cos(cosx)×sinx
C
sin(cosx)×(−sinx)
D
Both A and B are wrong; correct is −sinx cos(cosx)
73
If y = ln(tanx), dy/dx =
A
2/sin(2x)
B
1/tanx
C
sec²x/tanx
D
2csc(2x)
74
d/dx[f(2x+3)] = f'(2x+3) × ?
A
f(2x+3)
B
1
C
2
D
x+3
0
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