EGM211 Tutotial Sheet 1
Dierentiation Techniques
P. Musonda (Mr)
dy
Find f 0(x) or y0(x) or dx
:
1 First Principle
(ii)
f (x) = x3 − x + 1
√
f (x) = x
(iii)
f (x) = x2 − x − 6
(i)
2 General dierentiation
(i)
(ii)
y=
y = 5 sin 3x − cos 4x
(iii)
y=
(iv)
=
(v)
√
1− x
x
2
x2
y=
3
e5x
− 2 ln 2x − cos 5x − e3x
3
x3
−
x+1
√
x
+ 4x2.5
3 Product Rule
(i)
(ii)
y = e3x cos 4x
√
y = x3 ln 3x
(iii)
y = ex ln x cos x
(iv)
=
sin
√ 4x
x5
1
4 Quotient Rule
(i)
y=
2 cos 3x
x3
(ii)
y=
ln 2x
2x2 +3
(iii)
2xe4x
sin x
(iv)
y=
(v)
ln 5x
ln 6x
cot ax
5 Chain Rule
(i)
(ii)
y = 4 sin(x2 − 3)
y = 2 tan(5t2 − 2)
2 −4
(iii)
y = e3x
(iv)
y = ln(sin 3x2 )
(v)
(x ln x + 4)5
6 Implicit dierentiation
(i)
(ii)
3xy 2 + cos y 2 = 2x3 + 5
x − cos x2 +
y2
x
+ 3x5 = 4x3
(iii)
tan 5y − y ln x = 9 − 3x2 y
(iv)
exy − sin 8x = ln(xy) − 7
7 Logarithmic diereniation
(i)
(ii)
(iii)
(iv)
y = (ln x)x
(x2 − 2x)ln(x−3)
y = (x + 1)x+3
√
y = xe2 (x3 + x)
2
(v)
y=
x2 x sin x
cos 2x
(vi)
y=
(x+5)10 (3x−4)2
(3−x)(x+4)5
8 Parametric dierentiation
(i)
x = sin at
(ii)
y = 3t2 et
(iii)
x = te−t
and
where
a, b ∈ R.
x = ln(t4 )
and
and
y = cos bt,
y = 2t2 + 1
9 Inverse trig functions
(i)
y = sin−1 ( xy )
(ii)
y = cot−1 (ax)
(iii)
y = sec−1 (ax)
(iv)
y = csc−1 (ax)
10 Higher order Derivatives
Obtain the 3rd order derivative of
(i)
(ii)
y
with respect to
y = sin 3x
y = cos t
and
x = sin t
(iii)
y = ln 2x cos 3x
(iv)
y = sin−1 x
3
x.