5.2 Differentiation Rules

Let `f:x↦e^sinx.`

(a) Find `f^'(x)`

There is a point of inflexion on the graph of `f`, for `0 < x < 1`.

(b) Write down, but do not solve, an equation in terms of `x`, that would allow you to find the value of `x` at this point of inflexion.

(a) Given `f(x)=e^sinx`

Then `f^' (x)=cosx×e^sinx`

(b) `f^('')(x)=cos^2 x×e^sinx-sinx×e^sinx`

                   `=e^sinx (cos^2 x-sinx)`

For the point of inflexion, put `f^('')(x)=0`

`⇒ e^sinx (cos^2 x-sinx)=0" "` (or equivalent)

For the function `f:x↦x^2 ln⁡x,x>0`, find the function `f'`, the derivative of `f` with respect to `x`.

`f(x)=x^2 lnx`

`f^'(x)=2xlnx+x^2 (1/x)`

          `=2xlnx+x`

`f^':x↦2xlnx+x`

For the function `f:x↦1/2 sin2x+cosx`, find the possible values of `sinx` for which `f^' (x)=0`.

`f(x)=1/2 sin2x+cosx`

`f^' (x)=cos2x-sinx`

         `=1-2sin^2 x-sinx`

         `=(1+sinx)(1-2sinx)`

         `=0" when " sinx=-1" or " 1/2`  

Use l'Hôpital's rule to find `lim_(x→0) ((arctan2x)/(tan3x))`

attempt to differentiate numerator and denominator

`lim_(x→0) ((arctan2x)/(tan3x))`

`=lim_(x→0) ((2/(1+4x^2 )))/(3sec^2 3x)`

attempt to substitute `x=0`

`=2/3`

Given that `y=xe^(3x)+lnx`, find `dy/dx` and `(d^2 y)/( dx^2 ).`.

`y=xe^(3x)+lnx`

`⇒( dy)/( dx)=e^(3x)+3xe^(3x)+1/x`

`⇒(d^2 y)/( dx^2 )=3e^(3x)+3[e^(3x)+3xe^(3x) ]-1/x^2 =[6+9x]e^(3x)-1/x^2`

Let `y=sin^2 θ,0 ≤ θ ≤ π.`

(a) Find `dy/( dθ).`

(b) Hence find the values of `θ` for which `dy/( dθ)=2y`.

(a) attempt at chain rule or product rule

`dy/( dθ)=2sinθcosθ`

(b) `2sinθcosθ=2sin^2 θ`

`sinθ=0`

`θ=0,π`

obtaining `cosθ=sinθ`

`tanθ=1`

`θ=π/4`

Joon is a keen surfer and wants to model waves passing a particular point `P`, which is off the shore of his favourite beach. Joon sets up a model of the waves in terms of `h(t)`, the height of the water in metres, and `t`, the time in seconds from when he begins recording the height of the water at point `P`.

The function has the form `h(t)=pcos(π/6 t)+q,t ≥ 0.`

(a) Find the values of `p` and `q`.

(b) Find

(i) `h^'(t)`

(ii) `h^('')(t)`

Joon will begin to surf the wave when the rate of change of `h` with respect to `t`, at `P`, is at its maximum. This will first occur when `t=k`.

(c) (i) Find the value of `k`.

(ii) Find the height of the water at this time.

(a) `p=1.5;q=2`

(b) attempt at using chain rule 

(i) `h^' (t)=-π/4 sin(π/6 t)(=-0.785sin(π/6 t))`

(ii) `h^('')(t)=-π^2/24 cos(π/6 t)(=-0.411233…cos(π/6 t))`

(c)

(i) attempt to locate points of inflexion or max value of `h^' (t)`

`h^('')(t)=-π^2/24 cos(π/6 t)=0`

OR sketch on graph OR `t=3`OR `π/6 k=3π/2`

                                                          `k=9`.

(ii)  `h(9) = 2 (m)`

Use I'Hôpital's rule to find `lim(x→0) ((tan3x-3tanx)/(sin3x-3sinx))`

`=lim(x→0) ((3/(cos^2 3x)-3/(cos^2 x))/(3cos3x-3cosx))`

`=lim(x→0) ((cos^2 x-cos^2 3x)/(cos^2 3xcos^2 x(cos3x-cosx)))`

`=lim(x→0) ((cosx+cos3x)/(-cos^2 3xcos^2 x))`

attempt to substitute `x=0`

`=2/(-1)`

`=-2`

Use l'Hôpital's rule to find `lim(x→0)(sec^4 x-cos^2 x)/(x^4-x^2 ).`. 

`lim(x→0)(4sec^4 xtanx+2sinxcosx)/(4x^3-2x)`

`=lim(x→0)(16sec^4 xtan^2 x+4sec^6 x-2sin^2 x+2cos^2 x)/(12x^2-2)`

`=-3`

Use l'Hôpital's rule to find `lim_(x→1)(cos(x^2-1)-1)/(e^(x-1)-x)`

attempt to use l'Hôpital's rule

`=lim_(x→1)(-2xsin(x^2-1))/(e^(x-1)-1)`

substitution of 1 into their expression

`=0/0` hence use l'Hôpital's rule again

attempt to use product rule in numerator

`=lim_(x→1)(-4x^2 cos(x^2-1)-2sin(x^2-1))/e^(x-1)`

`=-4`