Differentiate with respect to `x`:
Let `y=sin x/(x + 1)`. Find `dy/dx `. (2 marks)
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Differentiate with respect to `x`:
Let `y=sin x/(x + 1)`. Find `dy/dx `. (2 marks)
`dy/dx = {cos x (x + 1) – sin x} / (x + 1)^2`
`y = sinx/(x + 1)`
`d/dx (u/v) = (u prime v – uv prime)/v^2`
`u` | `= sin x` | `v` | `= x + 1` |
`u prime` | `= cos x` | `\ \ \ v prime` | `= 1` |
`:.dy/dx = {cos x (x + 1) – sin x} / (x + 1)^2`
Let `y=xsinx.` Evaluate `dy/dx` for `x=pi.` (3 marks)
`- pi`
`y = x sin x`
`(dy)/(dx)` | `= x xx d/(dx) (sin x) + d/(dx) (x) xx sin x` |
`= x cos x + sin x` |
`text(When)\ \ x = pi,`
`(dy)/(dx)` | `= pi xx cos pi + sin pi` |
`= pi (-1) + 0` | |
`= – pi` |
Let `y = x^2 sin(x)`.
Find `(dy)/(dx)`. (1 mark)
`2x sin(x) + x^2 cosx`
`(dy)/(dx) = 2x sin(x) + x^2 cosx`
Let `g: R text(\ {−1}) -> R,\ \ g(x) = (sin(pi x))/(x + 1)`.
Evaluate `g prime(1)`. (2 marks)
`-pi/2`
`u = sin(pi x)` | `v = x + 1` | |
`u prime = pi cos(pi x)` | `v prime = 1` |
`g prime(x)` | `= (v u prime – u v prime)/v^2` |
`= ((x + 1) ⋅ pi cos(pi x) – sin (pi x))/(x + 1)^2` | |
`g prime(1)` | `= (2 pi cos(pi) – sin(pi))/2^2` |
`= (2 pi(-1) – 0)/4` | |
`= -pi/2` |
If `g(x) = x^2 sin (2x)`, find `g prime (pi/6).` (2 marks)
`(sqrt 3 pi)/6 + pi^2/36`
`g(x) = x^2 sin (2x)`
`text(Using Product Rule:)`
`(fh)′` | `= f′ h + fh′` |
`g prime (x)` | `= 2 x sin (2x) + 2x^2 cos (2x)` |
`:. g prime (pi/6)` | `= 2 (pi/6) sin (pi/3) + 2 (pi/6)^2 cos (pi/3)` |
`= pi/3 xx sqrt 3/2 + pi^2/18 xx 1/2` | |
`= (sqrt 3 pi)/6 + pi^2/36` |
If `f(x) = x/(sin(x))`, find `f prime (pi/2).` (2 marks)
`1`
`text(Using Quotient Rule:)`
`(h/g)′` | `= (h′ g – h g′)/g^2` |
`f prime (x)` | `= (1 xx sin (x) – x cos (x))/(sin x)^2` |
`:. f prime (pi/2)` | `= (sin (pi/2) – pi/2 xx cos (pi/2))/(sin(pi/2))^2` |
`= (1 – 0)/1^2` | |
`= 1` |
If `f(x)= 2 sin 3x - 3 tan x`, find `f′(0)`. (2 marks)
`3`
`y` | `= 2 sin 3x – 3 tan x` |
`(dy)/(dx)` | `= 6 cos 3x – 3 sec^2 x` |
`text(When)\ \ x = 0,`
`(dy)/(dx)` | `= 6 cos 0 – 3 sec^2 0` |
`= 6 (1) – 3/(cos^2 0)` | |
`= 6 – 3` | |
`= 3` |
Let `f(x) = (x^3)/(sin(x))`. Find `f′(x)`. (2 marks)
`(3x^2sin(x) – x^3cos(x))/(sin^2(x))`
`f(x) = (x^3)/(sin(x))`
`text(Using Quotient Rule:)`
`d/(dx)(u/v) = (vu′ – uv′)/(v^2)`
`:. f′(x) = (3x^2sin(x) – x^3cos(x))/(sin^2(x))`
If `y = x^2sin(x)`, find `(dy)/(dx)`. (2 marks)
`2xsin(x) + x^2cos(x)`
`text(Using product rule:)`
`(fg)′` | `= f′g + fg′` |
`:. (dy)/(dx)` | `= 2xsin(x) + x^2cos(x)` |