This page practises straightforward differentiation, chain rule and product rule.
I do not claim this is easy, and will provide good revision for C3 & C4 candidates
1 d ( 7x6 – 4x-3 )
dx
2 d ( 3x0.3 + 4x -0.4)
dx
3 d ( 3x⁴ – 4x) ³
dx
4 d ( 4 ( 23x – 4) ³ )
dx
5 d 4 .
dx (5x – 3) ²
6 d ( sin 5x )
dx
7 d ( 6x⁵ )
dt
8 d ( 6t³ + 14y )
dx
9 d 7(x6 – 4x) -3
dx
10 d (3x0.3 - 4x) -0.4
dx
11 d ( 3√x – 4 ³√x) 0.5
dx
12 d ( 4x-3) ( 23x³ – 4x)
dx
13 d 4 x .
dx (5x – 3)
14 d ( x sin x )
dx
15 d ( x³ sin x )
dx
16 d ( sin 3 x )
dx
17 d ( sin 3 x)-1 is also cosec 3x
dx
18 d ( e -0.4)
dx
19 d ( e 3x )
dx
20 d ( 4 e 3x - 2 )
dx
21 d ( 7x6 – 4e-3x )
dx
22 d ( 3e0.3x + 4e -0.4x)
dx
23 d ( 3x⁴ – 4ex) ³
dx
24 d ( 23e-x – 4x) ³
dx
25 d 4x²(5x³– 3)-12
dx
26 d ( sin 5x³ )
dx
27 d ( ø e -ø )
dø
28 d (ø e -ø )
dø
Differentiation in context: finding the stationary values of a function
29 A box is built from a square sheet of side a. Square corners of side x are cut out and the result is folded and sealed.
a) Show that the box has volume V=(a-2x).(a-2x).x
b) By finding the stationary values of V, find the maximum volume of the box
c) Show that the maximum surface area for the box occurs when it is a cube.
30 A ball flies through the air according to the function
2U² y = xT – 5 x² (1+T²) where U and T are constants for each flight.
a) Find where dy/dx = 0, which is at the top of the flight, and check that this is half the range (the larger x-value that makes y=0).
b) Treat 2U² y = xT – 5 x² (1+T²) as a quadratic in x and write the discriminant (the b² – 4ac part). For equal roots, the discriminant must be zero. Find the value of T when U=12 and y = 3.
c) Treat 2U² y = xT – 5 x² (1+T²) as a quadratic in T and write the discriminant. For U=12 and y=3, find the x-value that gives equal roots.
d) If y=0 and you consider x as a (product) function in T, show that T = ±1 gives the maximum.
