MA1S11 Tutorial Sheet 61
17 November 2015
Useful facts:
• Constant: If f = c a constant f 0 = 0.
• x to the r: If f (x) = xr then f 0 (x) = rxr−1 .
• Trigonometric functions
d
cos x = − sin x
dx
d
sin x = cos x
dx
• Product rule:
dg
df
d
(f g) = f
+ g
dx
dx dx
(1)
(2)
• Quotient rule:
d
dx
f
=
g
df
g
dx
dg
− f dx
g2
(3)
• Chain rule: With u = g(x)
df du
d
f (g(x)) =
= f 0 (u)g 0 (x).
dx
du dx
(4)
• Implicit differentiation: Example: find dy/dx for y 2 +x2 = 1, this gives a relationship between x and y without expressing y explicitly as a function of x. Differentiate
across
d 2
d
(x + y 2 ) =
1=0
(5)
dx
dx
so
dx dy 2
2x +
=0
(6)
dx
dx
and hence
dy
2x + 2y
=0
(7)
dx
or, solving for dy/dx
dy
x
=−
(8)
dx
y
1
Stefan Sint, sint@maths.tcd.ie, see also http://www.maths.tcd.ie/~sint/MA1S11.html
1
Questions
The numbers in brackets give the numbers of marks available for the question.
1. (2) Differentiate f (x) = x−n (with n a positive integer number) in 3 different ways:
• power rule for negative integer exponent
• quotient rule for 1/f (x) and the power rule for postive exponent.
• product rule for xn f (x) and power rule for positive exponent.
2. (4) Differentiate the following functions with respect to x
x3 cos 3x,
sin (cos x),
sin x
,
cos2 x
√
−x2 cos x
(9)
3. (2) Find the slope, that is dy/dx, of the curve
3y 2 − 2x2 = xy
(10)
at the point (1, 1).
Extra Questions
The questions are extra; you don’t need to do them in the tutorial class.
1. Find dy/dx for
3yx6 − 4x2 + 6 sin y 4 = 0
(11)
2. Derive the quotient rule for (f /g)0 from the product rule for (f h)0 with h = 1/g, and
the chain rule for h0 .
2