DIFFERENTIATION 1: FIRST RESULTS
(x0 )
df
5 minute review. Recall the definition dx
(x0 ) = limh→0 f (x0 +h)−f
by working
h
out the gradient of the line on the curve between (x0 , f (x0 )) and (x0 +h, f (x0 +h)).
Class warm-up. Find the derivatives of some of the following functions.
f (x) = x3 ; g(x) = x4 ; h(x) = xn ; j(x) =
1
;
x2
k(x) =
1
xn .
Problems. Choose from the below.
√
1. Differentiating x. Use the binomial theorem to show that
p
x0 + h ≈
√
√
x0 + h/(2 x0 )
when h x0 . Use the definition above to find the derivative of
√
x.
2. Differentiating tan x. Using the addition formulae for sin and cos, write
down an expression for tan(A + B) in terms of sines and cosines of A and B.
Put A = x0 and B = h, where h is very small, so that sin h ≈ h and cos h ≈ 1.
Deduce that
tan(x0 + h) ≈
tan x0 + h
.
1 − h tan x0
Hence work out the derivative of tan x at x = x0 .
3. The natural logarithm? . Put f (x) = ax . Recall from the videos that
ah − 1
f (x0 + h) − f (x0 )
= ax0 lim
,
h→0
h→0
h
h
lim
and that limh→0
ah −1
h
turns out to be ln(a). Let’s check this.
h
(a) First take a = 2. Use your calculator to work out a h−1 for h = 0.1, 0.01,
0.001 and 0.0001. How do these values compare to ln 2?
(b) Now let `(a) = limh→0
appears to be 1.
ah −1
h .
Use small values of h to check that `(e)
(ab)h − 1
ah − 1 bh − 1
= bh
+
and hence that `(ab) = `(a) +
h
h
h
`(b). This means that `(a) is a logarithm function to some base. Conclude by part (b) that the base is e.
(c) Show that
4. Some others. Differentiate the following functions, by any method whatsoever.
(a) x1000 , (b) x5 + x + 1, (c) e2x , (d) ln(2x), (e) sin(−5x).
DIFFERENTIATION 1: FIRST RESULTS
Selected answers and hints.
1
d √
1. As is well known, dx
( x) = 21 x− 2 .
2. The derivative of tan x is well known. . . use Google!
3. Covered in the video Standard Derivatives 2.
4. (a) 1000x999 , (b) 5x4 + 1, (c) 2e2x , (d) x1 , (e) −5 cos(−5x).
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