METHODS OF DIFFERENTIATION 291G
(c) Wheny=6ln2xthendy
dx= 6(
1
x)
=6
xProblem 8. Find the gradient of the curve
y= 3 x^4 − 2 x^2 + 5 x−2 at the points (0,−2) and
(1, 4).The gradient of a curve at a given point is given by
the corresponding value of the derivative. Thus, since
y= 3 x^4 − 2 x^2 + 5 x−2.
then the gradient=
dy
dx= 12 x^3 − 4 x+5.At the point (0,−2),x=0.
Thus the gradient=12(0)^3 −4(0)+ 5 = 5.
At the point (1, 4),x=1.
Thus the gradient=12(1)^3 −4(1)+ 5 = 13.
Problem 9. Determine the co-ordinates of the
point on the graphy= 3 x^2 − 7 x+2 where the
gradient is−1.The gradient of the curve is given by the derivative.
Wheny= 3 x^2 − 7 x+2 then
dy
dx= 6 x− 7Since the gradient is−1 then 6x− 7 =−1, from
which,x= 1
Whenx=1,y=3(1)^2 −7(1)+ 2 =− 2
Hence the gradient is−1 at the point (1,−2).
Now try the following exercise.
Exercise 117 Further problems on differen-
tiating common functionsIn Problems 1 to 6 find the differential coeffi-
cients of the given functions with respect to the
variable.- (a) 5x^5 (b) 2. 4 x^3.^5 (c)
1
x
[
(a) 25x^4 (b) 8. 4 x^2.^5 (c)−1
x^2]- (a)
− 4
x^2(b) 6 (c) 2x[
(a)8
x^3(b) 0 (c) 2]- (a) 2
√
x(b) 3√ 3
x^5 (c)4
√
x
[
(a)1
√
x(b) 5√ 3
x^2 (c)−2
√
x^3]- (a)
− 3√ (^3) x (b) (x−1)
(^2) (c) 2 sin 3x
⎡
⎣(a)
1
√ 3
x^4
(b) 2(x−1)
(c) 6 cos 3x
⎤
⎦
- (a)−4 cos 2x(b) 2e^6 x (c)
3
e^5 x
[
(a) 8 sin 2x(b) 12e^6 x(c)− 15
e^5 x]- (a) 4 ln 9x (b)
ex−e−x
2(c)1 −√
x⎡ x⎢
⎢
⎣(a)4
x(b)ex+e−x
2(c)− 1
x^2+12√
x^3⎤⎥
⎥
⎦- Find the gradient of the curvey= 2 t^4 +
3 t^3 −t+4 at the points (0, 4) and (1, 8).
[−1, 16] - Find the co-ordinates of the point on the
graphy= 5 x^2 − 3 x+1 where the gradient
is 2.
[( 1
2 ,3
4)]- (a) Differentiatey=
2
θ^2+2ln2θ−2 (cos 5θ+3 sin 2θ)−2
e^3 θ(b) Evaluatedy
dθin part (a) when θ=π
2,
correct to 4 significant figures.
⎡⎢
⎢
⎣(a)− 4
θ^3+2
θ+10 sin 5θ−12 cos 2θ+6
e^3 θ
(b) 22. 30⎤⎥
⎥
⎦- Evaluate
ds
dt, correct to 3 significant figures,whent=π
6givens=3 sint− 3 +√
t [3.29]